Properties

Label 4-1-1.1-c23e2-0-0
Degree $4$
Conductor $1$
Sign $1$
Analytic cond. $11.2361$
Root an. cond. $1.83085$
Motivic weight $23$
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  + 1.08e3·2-s + 3.39e5·3-s + 4.85e6·4-s + 7.30e7·5-s + 3.66e8·6-s − 1.35e9·7-s + 1.82e10·8-s − 5.40e10·9-s + 7.89e10·10-s + 8.56e11·11-s + 1.64e12·12-s + 4.37e12·13-s − 1.46e12·14-s + 2.48e13·15-s − 2.28e13·16-s + 2.54e14·17-s − 5.83e13·18-s + 4.26e12·19-s + 3.54e14·20-s − 4.61e14·21-s + 9.25e14·22-s − 8.14e15·23-s + 6.21e15·24-s − 1.51e16·25-s + 4.72e15·26-s − 4.38e16·27-s − 6.60e15·28-s + ⋯
L(s)  = 1  + 0.372·2-s + 1.10·3-s + 0.579·4-s + 0.669·5-s + 0.412·6-s − 0.259·7-s + 0.752·8-s − 0.573·9-s + 0.249·10-s + 0.905·11-s + 0.640·12-s + 0.677·13-s − 0.0968·14-s + 0.740·15-s − 0.325·16-s + 1.79·17-s − 0.213·18-s + 0.00839·19-s + 0.387·20-s − 0.287·21-s + 0.337·22-s − 1.78·23-s + 0.833·24-s − 1.27·25-s + 0.252·26-s − 1.51·27-s − 0.150·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(24-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+23/2)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(11.2361\)
Root analytic conductor: \(1.83085\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(3.576934756\)
\(L(\frac12)\) \(\approx\) \(3.576934756\)
\(L(\frac{25}{2})\) 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$D_{4}$ \( 1 - 135 p^{3} T - 3605 p^{10} T^{2} - 135 p^{26} T^{3} + p^{46} T^{4} \)
3$D_{4}$ \( 1 - 37720 p^{2} T + 77396530 p^{7} T^{2} - 37720 p^{25} T^{3} + p^{46} T^{4} \)
5$D_{4}$ \( 1 - 14613804 p T + 32768971378174 p^{4} T^{2} - 14613804 p^{24} T^{3} + p^{46} T^{4} \)
7$D_{4}$ \( 1 + 194169200 p T + 102109123349477250 p^{3} T^{2} + 194169200 p^{24} T^{3} + p^{46} T^{4} \)
11$D_{4}$ \( 1 - 77891088024 p T + \)\(16\!\cdots\!66\)\( p^{2} T^{2} - 77891088024 p^{24} T^{3} + p^{46} T^{4} \)
13$D_{4}$ \( 1 - 4376109322060 T + \)\(42\!\cdots\!90\)\( p T^{2} - 4376109322060 p^{23} T^{3} + p^{46} T^{4} \)
17$D_{4}$ \( 1 - 14942832211620 p T + \)\(15\!\cdots\!10\)\( p^{2} T^{2} - 14942832211620 p^{24} T^{3} + p^{46} T^{4} \)
19$D_{4}$ \( 1 - 224242156840 p T + \)\(34\!\cdots\!38\)\( p^{2} T^{2} - 224242156840 p^{24} T^{3} + p^{46} T^{4} \)
23$D_{4}$ \( 1 + 8144713079008560 T + \)\(58\!\cdots\!30\)\( T^{2} + 8144713079008560 p^{23} T^{3} + p^{46} T^{4} \)
29$D_{4}$ \( 1 - 20818433601623340 T + \)\(77\!\cdots\!78\)\( T^{2} - 20818433601623340 p^{23} T^{3} + p^{46} T^{4} \)
31$D_{4}$ \( 1 - 137714017177000384 T + \)\(40\!\cdots\!46\)\( T^{2} - 137714017177000384 p^{23} T^{3} + p^{46} T^{4} \)
37$D_{4}$ \( 1 + 897721264408967780 T + \)\(19\!\cdots\!70\)\( T^{2} + 897721264408967780 p^{23} T^{3} + p^{46} T^{4} \)
41$D_{4}$ \( 1 + 2294435477168314956 T + \)\(19\!\cdots\!26\)\( T^{2} + 2294435477168314956 p^{23} T^{3} + p^{46} T^{4} \)
43$D_{4}$ \( 1 + 1750760768619855800 T + \)\(73\!\cdots\!50\)\( T^{2} + 1750760768619855800 p^{23} T^{3} + p^{46} T^{4} \)
47$D_{4}$ \( 1 - 15759744217656780960 T + \)\(36\!\cdots\!10\)\( T^{2} - 15759744217656780960 p^{23} T^{3} + p^{46} T^{4} \)
53$D_{4}$ \( 1 + \)\(14\!\cdots\!20\)\( T + \)\(13\!\cdots\!10\)\( T^{2} + \)\(14\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
59$D_{4}$ \( 1 - \)\(28\!\cdots\!80\)\( T + \)\(10\!\cdots\!58\)\( T^{2} - \)\(28\!\cdots\!80\)\( p^{23} T^{3} + p^{46} T^{4} \)
61$D_{4}$ \( 1 + \)\(18\!\cdots\!36\)\( T + \)\(96\!\cdots\!86\)\( T^{2} + \)\(18\!\cdots\!36\)\( p^{23} T^{3} + p^{46} T^{4} \)
67$D_{4}$ \( 1 - \)\(17\!\cdots\!40\)\( T + \)\(24\!\cdots\!90\)\( T^{2} - \)\(17\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} \)
71$D_{4}$ \( 1 - \)\(30\!\cdots\!24\)\( T + \)\(93\!\cdots\!66\)\( T^{2} - \)\(30\!\cdots\!24\)\( p^{23} T^{3} + p^{46} T^{4} \)
73$D_{4}$ \( 1 + \)\(80\!\cdots\!60\)\( T + \)\(30\!\cdots\!30\)\( T^{2} + \)\(80\!\cdots\!60\)\( p^{23} T^{3} + p^{46} T^{4} \)
79$D_{4}$ \( 1 - \)\(62\!\cdots\!40\)\( T + \)\(47\!\cdots\!78\)\( T^{2} - \)\(62\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} \)
83$D_{4}$ \( 1 - \)\(68\!\cdots\!20\)\( T + \)\(16\!\cdots\!90\)\( T^{2} - \)\(68\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
89$D_{4}$ \( 1 - \)\(63\!\cdots\!20\)\( T + \)\(13\!\cdots\!38\)\( T^{2} - \)\(63\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
97$D_{4}$ \( 1 + \)\(31\!\cdots\!40\)\( T + \)\(53\!\cdots\!10\)\( T^{2} + \)\(31\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} \)
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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

−28.80701753675713985205512131895, −27.71975392964510574357267300963, −26.08002412254356619214664405113, −25.43480965722488614012096939507, −25.06262244345535832584628889273, −23.39018336851803561722438513387, −22.26073495304594102943229224580, −20.93411007176618065710844759575, −20.11436418542491146408551822951, −19.18449488041112370031013694544, −17.33046253304659494499678304849, −15.96477547641143977921582445988, −14.24685729890788642043577439014, −13.81955445804823991803132300494, −11.79514317423093929496864845919, −9.805490244224172864903671042701, −8.150088233572981620130000328332, −6.07058342244853426264178449878, −3.46542238195904954925506904775, −1.88602626271288641226254344200, 1.88602626271288641226254344200, 3.46542238195904954925506904775, 6.07058342244853426264178449878, 8.150088233572981620130000328332, 9.805490244224172864903671042701, 11.79514317423093929496864845919, 13.81955445804823991803132300494, 14.24685729890788642043577439014, 15.96477547641143977921582445988, 17.33046253304659494499678304849, 19.18449488041112370031013694544, 20.11436418542491146408551822951, 20.93411007176618065710844759575, 22.26073495304594102943229224580, 23.39018336851803561722438513387, 25.06262244345535832584628889273, 25.43480965722488614012096939507, 26.08002412254356619214664405113, 27.71975392964510574357267300963, 28.80701753675713985205512131895

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