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Properties

Label 4-74e2-1.1-c1e2-0-3
Degree $4$
Conductor $5476$
Sign $1$
Analytic cond. $0.349154$
Root an. cond. $0.768695$
Motivic weight $1$
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  + 2-s + 2·3-s − 3·5-s + 2·6-s + 4·7-s − 8-s + 3·9-s − 3·10-s − 12·11-s − 2·13-s + 4·14-s − 6·15-s − 16-s − 3·17-s + 3·18-s − 2·19-s + 8·21-s − 12·22-s + 12·23-s − 2·24-s + 5·25-s − 2·26-s + 10·27-s + 6·29-s − 6·30-s + 4·31-s − 24·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 1.34·5-s + 0.816·6-s + 1.51·7-s − 0.353·8-s + 9-s − 0.948·10-s − 3.61·11-s − 0.554·13-s + 1.06·14-s − 1.54·15-s − 1/4·16-s − 0.727·17-s + 0.707·18-s − 0.458·19-s + 1.74·21-s − 2.55·22-s + 2.50·23-s − 0.408·24-s + 25-s − 0.392·26-s + 1.92·27-s + 1.11·29-s − 1.09·30-s + 0.718·31-s − 4.17·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5476 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5476\)    =    \(2^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(0.349154\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5476,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.299289952\)
\(L(\frac12)\) \(\approx\) \(1.299289952\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 - T + T^{2} \)
37$C_2$ \( 1 - 11 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
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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

−14.95653443958464231055700798311, −14.46612310158116595310787215373, −13.70109124751105375943807488161, −13.15604242257151514758203168890, −12.92716226973792120455014388449, −12.45115738634829876929923446628, −11.33939819520528070453147229241, −11.26414665020741052750068267436, −10.40682458920880950055875691345, −10.08308586760505866467050837192, −8.766420929132927914421575909952, −8.243035394302099928905449408459, −8.165443290775452655722194638782, −7.47356788247230065397830730477, −6.89372335788240693191062248397, −5.25336430560591741725632318855, −4.68609982432455321402057069443, −4.62481559599403455827310685414, −2.84776382367424505759218468873, −2.79092090144508449658333842243, 2.79092090144508449658333842243, 2.84776382367424505759218468873, 4.62481559599403455827310685414, 4.68609982432455321402057069443, 5.25336430560591741725632318855, 6.89372335788240693191062248397, 7.47356788247230065397830730477, 8.165443290775452655722194638782, 8.243035394302099928905449408459, 8.766420929132927914421575909952, 10.08308586760505866467050837192, 10.40682458920880950055875691345, 11.26414665020741052750068267436, 11.33939819520528070453147229241, 12.45115738634829876929923446628, 12.92716226973792120455014388449, 13.15604242257151514758203168890, 13.70109124751105375943807488161, 14.46612310158116595310787215373, 14.95653443958464231055700798311

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