Properties

Label 4-637e2-1.1-c1e2-0-6
Degree $4$
Conductor $405769$
Sign $1$
Analytic cond. $25.8721$
Root an. cond. $2.25532$
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 − 6·9-s + 12·16-s + 2·23-s − 3·25-s − 10·29-s − 24·36-s + 18·43-s + 22·53-s + 32·64-s + 30·79-s + 27·81-s + 8·92-s − 12·100-s + 16·107-s − 38·113-s − 40·116-s + 22·121-s + 127-s + 131-s + 137-s + 139-s − 72·144-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 2·4-s − 2·9-s + 3·16-s + 0.417·23-s − 3/5·25-s − 1.85·29-s − 4·36-s + 2.74·43-s + 3.02·53-s + 4·64-s + 3.37·79-s + 3·81-s + 0.834·92-s − 6/5·100-s + 1.54·107-s − 3.57·113-s − 3.71·116-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(405769\)    =    \(7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(25.8721\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 405769,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.670297364\)
\(L(\frac12)\) \(\approx\) \(2.670297364\)
\(L(\frac{3}{2})\) 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)$
bad7 \( 1 \)
13$C_2$ \( 1 + p T^{2} \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 81 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 159 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 165 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 131 T^{2} + p^{2} T^{4} \)
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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

−10.95501419740165719230398927607, −10.53758706961148283263872825775, −10.17301053760386419033308267863, −9.310666724135831601117130001293, −9.177948290762569783891091012030, −8.599949348034413221895450657824, −7.87888024039073588277570836718, −7.82472286374025567874595526250, −7.26779433921307144355118806726, −6.80306429687961310874924384934, −6.24538692255915754878503141910, −5.85277215831893289814021448477, −5.54240797548505922390033308921, −5.18648095530453650072178156842, −3.93798235111331440903582746562, −3.60556027641073429976861898300, −2.90183652915865826561172861350, −2.36122142279252356623599827622, −2.11003093269110094512873523839, −0.865643589004246538352849698417, 0.865643589004246538352849698417, 2.11003093269110094512873523839, 2.36122142279252356623599827622, 2.90183652915865826561172861350, 3.60556027641073429976861898300, 3.93798235111331440903582746562, 5.18648095530453650072178156842, 5.54240797548505922390033308921, 5.85277215831893289814021448477, 6.24538692255915754878503141910, 6.80306429687961310874924384934, 7.26779433921307144355118806726, 7.82472286374025567874595526250, 7.87888024039073588277570836718, 8.599949348034413221895450657824, 9.177948290762569783891091012030, 9.310666724135831601117130001293, 10.17301053760386419033308267863, 10.53758706961148283263872825775, 10.95501419740165719230398927607

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