Properties

Label 4-1216e2-1.1-c1e2-0-0
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
Motivic weight $1$
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  + 3-s − 4·5-s + 3·9-s − 6·11-s + 2·13-s − 4·15-s − 2·17-s − 19-s − 6·23-s + 5·25-s + 8·27-s − 4·29-s − 20·31-s − 6·33-s − 4·37-s + 2·39-s − 9·41-s − 4·43-s − 12·45-s + 12·47-s − 14·49-s − 2·51-s − 2·53-s + 24·55-s − 57-s − 59-s − 8·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s + 9-s − 1.80·11-s + 0.554·13-s − 1.03·15-s − 0.485·17-s − 0.229·19-s − 1.25·23-s + 25-s + 1.53·27-s − 0.742·29-s − 3.59·31-s − 1.04·33-s − 0.657·37-s + 0.320·39-s − 1.40·41-s − 0.609·43-s − 1.78·45-s + 1.75·47-s − 2·49-s − 0.280·51-s − 0.274·53-s + 3.23·55-s − 0.132·57-s − 0.130·59-s − 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2352971243\)
\(L(\frac12)\) \(\approx\) \(0.2352971243\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 + T + p T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−10.42112824027713223787485408378, −9.279318677952284270476246114242, −9.194645270905816779697029808030, −8.627207374528811572045863895809, −7.998133238397861103457405460033, −7.88213138654611469930968810318, −7.82660739483147917959467981185, −7.14052847001835329244736612695, −6.83738908519454562562143914057, −6.35079965555136488708837351961, −5.56023829364467878995641976357, −5.12404206937091652157637415845, −4.88584097114635744909293674631, −3.92492264563539322797459055727, −3.89916287413110371298495766894, −3.53557239646990990256404256307, −2.85199342952786188851794984299, −2.07829031274344949334775058911, −1.66383371607342896001168565101, −0.19140794077643373687232908545, 0.19140794077643373687232908545, 1.66383371607342896001168565101, 2.07829031274344949334775058911, 2.85199342952786188851794984299, 3.53557239646990990256404256307, 3.89916287413110371298495766894, 3.92492264563539322797459055727, 4.88584097114635744909293674631, 5.12404206937091652157637415845, 5.56023829364467878995641976357, 6.35079965555136488708837351961, 6.83738908519454562562143914057, 7.14052847001835329244736612695, 7.82660739483147917959467981185, 7.88213138654611469930968810318, 7.998133238397861103457405460033, 8.627207374528811572045863895809, 9.194645270905816779697029808030, 9.279318677952284270476246114242, 10.42112824027713223787485408378

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