Properties

Degree $4$
Conductor $13505625$
Sign $1$
Motivic weight $1$
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 + 2·3-s + 4-s + 4·6-s + 3·9-s − 4·11-s + 2·12-s + 8·13-s + 16-s + 4·17-s + 6·18-s − 8·22-s + 4·23-s + 16·26-s + 4·27-s − 8·29-s + 8·31-s − 2·32-s − 8·33-s + 8·34-s + 3·36-s + 8·37-s + 16·39-s − 4·41-s − 4·44-s + 8·46-s + 2·48-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s + 9-s − 1.20·11-s + 0.577·12-s + 2.21·13-s + 1/4·16-s + 0.970·17-s + 1.41·18-s − 1.70·22-s + 0.834·23-s + 3.13·26-s + 0.769·27-s − 1.48·29-s + 1.43·31-s − 0.353·32-s − 1.39·33-s + 1.37·34-s + 1/2·36-s + 1.31·37-s + 2.56·39-s − 0.624·41-s − 0.603·44-s + 1.17·46-s + 0.288·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13505625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3675} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13505625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(10.89262573\)
\(L(\frac12)\) \(\approx\) \(10.89262573\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
7 \( 1 \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 168 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 64 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 208 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−8.518941975047728879078266854356, −8.274922688455764216118004818212, −7.999832316299798033765249200153, −7.73313934191243928199490154589, −7.27807335604176674189236749169, −6.82920622926100299214089502382, −6.39824163937052954811198460182, −5.91157177155520573146655747080, −5.67755741483716787720495935634, −5.27697974758996349470661069181, −4.82051074060462128697470921502, −4.49052024529529474131550125834, −3.96374543935467831819978162126, −3.75349692209605763541666174657, −3.16899483023366214997402616957, −3.15740738618990452251776045459, −2.49562865035449070337328342018, −1.90980432137990327391261256039, −1.32681827226441093528655579329, −0.75808932091979279254199746409, 0.75808932091979279254199746409, 1.32681827226441093528655579329, 1.90980432137990327391261256039, 2.49562865035449070337328342018, 3.15740738618990452251776045459, 3.16899483023366214997402616957, 3.75349692209605763541666174657, 3.96374543935467831819978162126, 4.49052024529529474131550125834, 4.82051074060462128697470921502, 5.27697974758996349470661069181, 5.67755741483716787720495935634, 5.91157177155520573146655747080, 6.39824163937052954811198460182, 6.82920622926100299214089502382, 7.27807335604176674189236749169, 7.73313934191243928199490154589, 7.999832316299798033765249200153, 8.274922688455764216118004818212, 8.518941975047728879078266854356

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