Properties

Label 4-74e2-1.1-c2e2-0-1
Degree $4$
Conductor $5476$
Sign $1$
Analytic cond. $4.06568$
Root an. cond. $1.41998$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 6·5-s − 8·7-s + 2·9-s + 12·10-s + 6·13-s − 16·14-s − 4·16-s + 46·17-s + 4·18-s + 20·19-s + 12·20-s − 20·23-s + 18·25-s + 12·26-s − 16·28-s − 38·29-s − 36·31-s − 8·32-s + 92·34-s − 48·35-s + 4·36-s + 40·38-s + 84·43-s + 12·45-s − 40·46-s + ⋯
L(s)  = 1  + 2-s + 1/2·4-s + 6/5·5-s − 8/7·7-s + 2/9·9-s + 6/5·10-s + 6/13·13-s − 8/7·14-s − 1/4·16-s + 2.70·17-s + 2/9·18-s + 1.05·19-s + 3/5·20-s − 0.869·23-s + 0.719·25-s + 6/13·26-s − 4/7·28-s − 1.31·29-s − 1.16·31-s − 1/4·32-s + 2.70·34-s − 1.37·35-s + 1/9·36-s + 1.05·38-s + 1.95·43-s + 4/15·45-s − 0.869·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5476 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5476\)    =    \(2^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(4.06568\)
Root analytic conductor: \(1.41998\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{74} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5476,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.656706294\)
\(L(\frac12)\) \(\approx\) \(2.656706294\)
\(L(2)\) 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$C_2$ \( 1 - p T + p T^{2} \)
37$C_2$ \( 1 + p^{2} T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{4} T^{4} \)
5$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2$ \( ( 1 + 4 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 226 T^{2} + p^{4} T^{4} \)
13$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2$ \( ( 1 - 30 T + p^{2} T^{2} )( 1 - 16 T + p^{2} T^{2} ) \)
19$C_2^2$ \( 1 - 20 T + 200 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} \)
23$C_2^2$ \( 1 + 20 T + 200 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2^2$ \( 1 + 38 T + 722 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \)
31$C_2^2$ \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2^2$ \( 1 + 2114 T^{2} + p^{4} T^{4} \)
43$C_2^2$ \( 1 - 84 T + 3528 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2$ \( ( 1 + 44 T + p^{2} T^{2} )^{2} \)
53$C_2$ \( ( 1 + 80 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 108 T + 5832 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 - 8834 T^{2} + p^{4} T^{4} \)
71$C_2$ \( ( 1 - 124 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 10558 T^{2} + p^{4} T^{4} \)
79$C_2^2$ \( 1 - 28 T + 392 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2$ \( ( 1 + 64 T + p^{2} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 34 T + 578 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 258 T + 33282 T^{2} - 258 p^{2} T^{3} + p^{4} T^{4} \)
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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

−14.74582601661900919183475078088, −13.94971302798630805091721976874, −13.67807641372599725004753761468, −12.80080056819426928365159945434, −12.65572488415663634665699921647, −12.28436777667639013895495109900, −11.29772222717272103930219538802, −10.84711011935171845950599526817, −9.776008111707046971156775444856, −9.680052062216690319669824061045, −9.401469888143138558411485371251, −8.032794725847557928015040339350, −7.56444323665446958086294095506, −6.61779079946939470987437219179, −5.83275302681272204421359771488, −5.76186677957676864048926210040, −4.84866898617643768664888855560, −3.37875699457624295053014397407, −3.33921956009323824632251740169, −1.65956295605923317123107545788, 1.65956295605923317123107545788, 3.33921956009323824632251740169, 3.37875699457624295053014397407, 4.84866898617643768664888855560, 5.76186677957676864048926210040, 5.83275302681272204421359771488, 6.61779079946939470987437219179, 7.56444323665446958086294095506, 8.032794725847557928015040339350, 9.401469888143138558411485371251, 9.680052062216690319669824061045, 9.776008111707046971156775444856, 10.84711011935171845950599526817, 11.29772222717272103930219538802, 12.28436777667639013895495109900, 12.65572488415663634665699921647, 12.80080056819426928365159945434, 13.67807641372599725004753761468, 13.94971302798630805091721976874, 14.74582601661900919183475078088

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