Properties

Label 4-39e4-1.1-c1e2-0-4
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $147.507$
Root an. cond. $3.48500$
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  − 4·4-s + 12·16-s − 10·25-s + 26·43-s + 13·49-s + 26·61-s − 32·64-s + 26·79-s + 40·100-s + 26·103-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 104·172-s + 173-s + 179-s + 181-s + 191-s + 193-s − 52·196-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s − 2·25-s + 3.96·43-s + 13/7·49-s + 3.32·61-s − 4·64-s + 2.92·79-s + 4·100-s + 2.56·103-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 7.92·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 3.71·196-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(147.507\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2313441,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.225233739\)
\(L(\frac12)\) \(\approx\) \(1.225233739\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
13 \( 1 \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 143 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 169 T^{2} + p^{2} T^{4} \)
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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

−9.539231180083738508171994864327, −9.278550982516379096278423471452, −8.822791374383340678835143040774, −8.728135637863489422975384731820, −8.016079466379376999417096621777, −7.72536714583701354336606708079, −7.57777450618659838656119570441, −6.90785819662818302222573672090, −6.20338026813584923889916123382, −5.86759524499834832980391948758, −5.42673022080874236713083437174, −5.24169675575209242293511872037, −4.48531479108800978823334132249, −4.20249564433908539379775542485, −3.72835706033966006853332815739, −3.59965430816273419096247497725, −2.54093920856658529073826340380, −2.15113593829640636363996300451, −0.998216983262098798233120834662, −0.58244129014888587342252076828, 0.58244129014888587342252076828, 0.998216983262098798233120834662, 2.15113593829640636363996300451, 2.54093920856658529073826340380, 3.59965430816273419096247497725, 3.72835706033966006853332815739, 4.20249564433908539379775542485, 4.48531479108800978823334132249, 5.24169675575209242293511872037, 5.42673022080874236713083437174, 5.86759524499834832980391948758, 6.20338026813584923889916123382, 6.90785819662818302222573672090, 7.57777450618659838656119570441, 7.72536714583701354336606708079, 8.016079466379376999417096621777, 8.728135637863489422975384731820, 8.822791374383340678835143040774, 9.278550982516379096278423471452, 9.539231180083738508171994864327

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