Properties

Label 4-1-1.1-c31e2-0-0
Degree $4$
Conductor $1$
Sign $1$
Analytic cond. $37.0602$
Root an. cond. $2.46732$
Motivic weight $31$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.99e4·2-s + 1.73e7·3-s − 4.62e8·4-s − 1.93e10·5-s + 6.93e11·6-s + 3.02e13·7-s − 1.49e13·8-s − 5.17e14·9-s − 7.74e14·10-s − 7.78e15·11-s − 8.03e15·12-s + 7.47e16·13-s + 1.20e18·14-s − 3.36e17·15-s − 8.28e17·16-s + 1.72e19·17-s − 2.06e19·18-s − 1.23e19·19-s + 8.97e18·20-s + 5.25e20·21-s − 3.10e20·22-s + 1.89e21·23-s − 2.60e20·24-s − 3.72e21·25-s + 2.98e21·26-s − 1.24e22·27-s − 1.40e22·28-s + ⋯
L(s)  = 1  + 0.862·2-s + 0.698·3-s − 0.215·4-s − 0.284·5-s + 0.602·6-s + 2.40·7-s − 0.150·8-s − 0.837·9-s − 0.245·10-s − 0.561·11-s − 0.150·12-s + 0.404·13-s + 2.07·14-s − 0.198·15-s − 0.179·16-s + 1.45·17-s − 0.722·18-s − 0.186·19-s + 0.0612·20-s + 1.68·21-s − 0.484·22-s + 1.48·23-s − 0.105·24-s − 0.800·25-s + 0.349·26-s − 0.813·27-s − 0.519·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(32-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+31/2)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(37.0602\)
Root analytic conductor: \(2.46732\)
Motivic weight: \(31\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1,\ (\ :31/2, 31/2),\ 1)\)

Particular Values

\(L(16)\) \(\approx\) \(4.421604976\)
\(L(\frac12)\) \(\approx\) \(4.421604976\)
\(L(\frac{33}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
good2$D_{4}$ \( 1 - 4995 p^{3} T + 2011345 p^{10} T^{2} - 4995 p^{34} T^{3} + p^{62} T^{4} \)
3$D_{4}$ \( 1 - 214360 p^{4} T + 13870620790 p^{10} T^{2} - 214360 p^{35} T^{3} + p^{62} T^{4} \)
5$D_{4}$ \( 1 + 3878243604 p T + 1313821413051106982 p^{5} T^{2} + 3878243604 p^{32} T^{3} + p^{62} T^{4} \)
7$D_{4}$ \( 1 - 617500562800 p^{2} T + \)\(31\!\cdots\!50\)\( p^{5} T^{2} - 617500562800 p^{33} T^{3} + p^{62} T^{4} \)
11$D_{4}$ \( 1 + 7782353745118776 T + \)\(32\!\cdots\!46\)\( p^{2} T^{2} + 7782353745118776 p^{31} T^{3} + p^{62} T^{4} \)
13$D_{4}$ \( 1 - 5746842542327740 p T + \)\(55\!\cdots\!10\)\( p^{3} T^{2} - 5746842542327740 p^{32} T^{3} + p^{62} T^{4} \)
17$D_{4}$ \( 1 - 17224607828987089380 T + \)\(20\!\cdots\!70\)\( p T^{2} - 17224607828987089380 p^{31} T^{3} + p^{62} T^{4} \)
19$D_{4}$ \( 1 + 651082280422219160 p T + \)\(24\!\cdots\!58\)\( p^{2} T^{2} + 651082280422219160 p^{32} T^{3} + p^{62} T^{4} \)
23$D_{4}$ \( 1 - 82493253999561357360 p T + \)\(61\!\cdots\!70\)\( p^{2} T^{2} - 82493253999561357360 p^{32} T^{3} + p^{62} T^{4} \)
29$D_{4}$ \( 1 - \)\(44\!\cdots\!60\)\( p T + \)\(98\!\cdots\!38\)\( p^{2} T^{2} - \)\(44\!\cdots\!60\)\( p^{32} T^{3} + p^{62} T^{4} \)
31$D_{4}$ \( 1 - \)\(12\!\cdots\!64\)\( T + \)\(22\!\cdots\!86\)\( T^{2} - \)\(12\!\cdots\!64\)\( p^{31} T^{3} + p^{62} T^{4} \)
37$D_{4}$ \( 1 + \)\(83\!\cdots\!60\)\( T + \)\(56\!\cdots\!70\)\( T^{2} + \)\(83\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \)
41$D_{4}$ \( 1 - \)\(87\!\cdots\!84\)\( T + \)\(12\!\cdots\!46\)\( T^{2} - \)\(87\!\cdots\!84\)\( p^{31} T^{3} + p^{62} T^{4} \)
43$D_{4}$ \( 1 + \)\(18\!\cdots\!00\)\( T + \)\(64\!\cdots\!50\)\( T^{2} + \)\(18\!\cdots\!00\)\( p^{31} T^{3} + p^{62} T^{4} \)
47$D_{4}$ \( 1 - \)\(95\!\cdots\!20\)\( T + \)\(15\!\cdots\!10\)\( T^{2} - \)\(95\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \)
53$D_{4}$ \( 1 - \)\(19\!\cdots\!60\)\( T + \)\(57\!\cdots\!10\)\( T^{2} - \)\(19\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \)
59$D_{4}$ \( 1 + \)\(19\!\cdots\!20\)\( T + \)\(10\!\cdots\!18\)\( T^{2} + \)\(19\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \)
61$D_{4}$ \( 1 + \)\(12\!\cdots\!76\)\( T + \)\(80\!\cdots\!66\)\( T^{2} + \)\(12\!\cdots\!76\)\( p^{31} T^{3} + p^{62} T^{4} \)
67$D_{4}$ \( 1 + \)\(96\!\cdots\!20\)\( T + \)\(81\!\cdots\!90\)\( T^{2} + \)\(96\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \)
71$D_{4}$ \( 1 - \)\(55\!\cdots\!44\)\( T + \)\(32\!\cdots\!26\)\( T^{2} - \)\(55\!\cdots\!44\)\( p^{31} T^{3} + p^{62} T^{4} \)
73$D_{4}$ \( 1 - \)\(62\!\cdots\!80\)\( T + \)\(12\!\cdots\!30\)\( T^{2} - \)\(62\!\cdots\!80\)\( p^{31} T^{3} + p^{62} T^{4} \)
79$D_{4}$ \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(80\!\cdots\!58\)\( T^{2} + \)\(11\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \)
83$D_{4}$ \( 1 + \)\(26\!\cdots\!60\)\( T + \)\(53\!\cdots\!90\)\( T^{2} + \)\(26\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \)
89$D_{4}$ \( 1 + \)\(21\!\cdots\!80\)\( T + \)\(47\!\cdots\!78\)\( T^{2} + \)\(21\!\cdots\!80\)\( p^{31} T^{3} + p^{62} T^{4} \)
97$D_{4}$ \( 1 + \)\(90\!\cdots\!80\)\( T + \)\(97\!\cdots\!10\)\( T^{2} + \)\(90\!\cdots\!80\)\( p^{31} T^{3} + p^{62} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.59815045482878068994052177070, −24.74601604359586371830940866206, −23.33111690817342398940431907316, −23.23077440410111646454398569488, −21.15899146928555972189445545945, −21.11130247220277519068551323863, −19.67496808881181391172802518323, −18.21187401009027052607053012703, −17.21085878815702513924416483796, −15.25153064420284295903901090958, −14.18215877875086330379784124974, −13.90495256268892525256775977574, −12.00980514724768876514313786268, −10.84651953735742116019517462645, −8.539617187885902554201134298215, −7.899175154495250600673870691710, −5.28996082164204900634330635759, −4.48743998605134772879277686601, −2.85476292460552992262642155636, −1.22818716050636831349375206415, 1.22818716050636831349375206415, 2.85476292460552992262642155636, 4.48743998605134772879277686601, 5.28996082164204900634330635759, 7.899175154495250600673870691710, 8.539617187885902554201134298215, 10.84651953735742116019517462645, 12.00980514724768876514313786268, 13.90495256268892525256775977574, 14.18215877875086330379784124974, 15.25153064420284295903901090958, 17.21085878815702513924416483796, 18.21187401009027052607053012703, 19.67496808881181391172802518323, 21.11130247220277519068551323863, 21.15899146928555972189445545945, 23.23077440410111646454398569488, 23.33111690817342398940431907316, 24.74601604359586371830940866206, 25.59815045482878068994052177070

Graph of the $Z$-function along the critical line