Properties

Label 4-1134e2-1.1-c1e2-0-37
Degree $4$
Conductor $1285956$
Sign $1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 2·7-s + 3·11-s + 16-s − 3·23-s + 2·25-s + 2·28-s + 3·29-s − 11·37-s + 13·43-s + 3·44-s − 3·49-s + 21·53-s + 64-s − 5·67-s + 6·71-s + 6·77-s − 14·79-s − 3·92-s + 2·100-s + 3·107-s + 10·109-s + 2·112-s + 3·116-s − 13·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.755·7-s + 0.904·11-s + 1/4·16-s − 0.625·23-s + 2/5·25-s + 0.377·28-s + 0.557·29-s − 1.80·37-s + 1.98·43-s + 0.452·44-s − 3/7·49-s + 2.88·53-s + 1/8·64-s − 0.610·67-s + 0.712·71-s + 0.683·77-s − 1.57·79-s − 0.312·92-s + 1/5·100-s + 0.290·107-s + 0.957·109-s + 0.188·112-s + 0.278·116-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.996840572\)
\(L(\frac12)\) \(\approx\) \(2.996840572\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \)
31$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 139 T^{2} + p^{2} T^{4} \)
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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

−8.086957378909260827507725498113, −7.36183531325089434705031299939, −7.12751071510912888473626543776, −6.88293266829974904231241315122, −6.11537814027800075015041152104, −5.91546075633974453763631173072, −5.37028948666120768681653191605, −4.87038393691364837970879749279, −4.30084316977977013470125198039, −3.93637597882520339189729774411, −3.37114565437925398598148669121, −2.69464080839549040218065639632, −2.11200230426453434596424916503, −1.54613197909600252293549042817, −0.799877307541708731708021076720, 0.799877307541708731708021076720, 1.54613197909600252293549042817, 2.11200230426453434596424916503, 2.69464080839549040218065639632, 3.37114565437925398598148669121, 3.93637597882520339189729774411, 4.30084316977977013470125198039, 4.87038393691364837970879749279, 5.37028948666120768681653191605, 5.91546075633974453763631173072, 6.11537814027800075015041152104, 6.88293266829974904231241315122, 7.12751071510912888473626543776, 7.36183531325089434705031299939, 8.086957378909260827507725498113

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