Properties

Degree $4$
Conductor $13505625$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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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: \(2\)
Selberg data: \((4,\ 13505625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−8.147780811848624937742704656157, −7.63571116907737384794707990109, −7.49481170028502263168271760327, −7.25024574565515031196902629764, −6.81005579146774916998539383137, −6.38982067264027061216941258622, −5.74458253172374612675038724456, −5.65490291255029523785641071670, −5.20990385709902739148000490349, −4.98299574076440590542957319104, −4.56904515637746052803139740964, −4.46161838387592170608390615408, −3.68889164352743747936990578626, −3.59974204500420010922240062161, −2.63970184926475802301233401722, −2.44105420011153060180056091389, −1.98895692884713106185784380151, −1.10842947150966580383950259716, 0, 0, 1.10842947150966580383950259716, 1.98895692884713106185784380151, 2.44105420011153060180056091389, 2.63970184926475802301233401722, 3.59974204500420010922240062161, 3.68889164352743747936990578626, 4.46161838387592170608390615408, 4.56904515637746052803139740964, 4.98299574076440590542957319104, 5.20990385709902739148000490349, 5.65490291255029523785641071670, 5.74458253172374612675038724456, 6.38982067264027061216941258622, 6.81005579146774916998539383137, 7.25024574565515031196902629764, 7.49481170028502263168271760327, 7.63571116907737384794707990109, 8.147780811848624937742704656157

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