Properties

Label 4-637e2-1.1-c1e2-0-5
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  − 2-s + 2·4-s − 5·8-s + 3·9-s + 3·11-s + 2·13-s + 5·16-s + 7·17-s − 3·18-s − 7·19-s − 3·22-s + 6·23-s + 5·25-s − 2·26-s − 10·29-s − 10·32-s − 7·34-s + 6·36-s − 8·37-s + 7·38-s + 4·43-s + 6·44-s − 6·46-s + 7·47-s − 5·50-s + 4·52-s + 3·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 4-s − 1.76·8-s + 9-s + 0.904·11-s + 0.554·13-s + 5/4·16-s + 1.69·17-s − 0.707·18-s − 1.60·19-s − 0.639·22-s + 1.25·23-s + 25-s − 0.392·26-s − 1.85·29-s − 1.76·32-s − 1.20·34-s + 36-s − 1.31·37-s + 1.13·38-s + 0.609·43-s + 0.904·44-s − 0.884·46-s + 1.02·47-s − 0.707·50-s + 0.554·52-s + 0.412·53-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

\(L(1)\) \(\approx\) \(1.651111338\)
\(L(\frac12)\) \(\approx\) \(1.651111338\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−10.68458612053644865753977119388, −10.52744305472162628133655378335, −9.882074596398518959537477103817, −9.383676610850082375905670500794, −8.978064617523603637430001594600, −8.905146449742066753527586387736, −8.246256936157764178432145919971, −7.65220415425088187959310737100, −7.25806605948748998293713394908, −6.80102784244485052352179526989, −6.54522585179525654570302498294, −5.77211743066663245821371098304, −5.70699944540611899821201334029, −4.81098966747532589402369096897, −4.14720759242437579204427323377, −3.37986883239747882751581002366, −3.30887562154071612851098618355, −2.23190411749249809139188603495, −1.64588763118373417066028485251, −0.831276927847173280375774686574, 0.831276927847173280375774686574, 1.64588763118373417066028485251, 2.23190411749249809139188603495, 3.30887562154071612851098618355, 3.37986883239747882751581002366, 4.14720759242437579204427323377, 4.81098966747532589402369096897, 5.70699944540611899821201334029, 5.77211743066663245821371098304, 6.54522585179525654570302498294, 6.80102784244485052352179526989, 7.25806605948748998293713394908, 7.65220415425088187959310737100, 8.246256936157764178432145919971, 8.905146449742066753527586387736, 8.978064617523603637430001594600, 9.383676610850082375905670500794, 9.882074596398518959537477103817, 10.52744305472162628133655378335, 10.68458612053644865753977119388

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