Properties

Label 4-637e2-1.1-c1e2-0-4
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  − 3·2-s + 2·3-s + 4·4-s + 3·5-s − 6·6-s − 3·8-s − 3·9-s − 9·10-s + 8·12-s + 2·13-s + 6·15-s + 3·16-s + 6·17-s + 9·18-s + 12·20-s − 6·24-s + 25-s − 6·26-s − 14·27-s − 3·29-s − 18·30-s − 3·31-s − 6·32-s − 18·34-s − 12·36-s + 4·39-s − 9·40-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.15·3-s + 2·4-s + 1.34·5-s − 2.44·6-s − 1.06·8-s − 9-s − 2.84·10-s + 2.30·12-s + 0.554·13-s + 1.54·15-s + 3/4·16-s + 1.45·17-s + 2.12·18-s + 2.68·20-s − 1.22·24-s + 1/5·25-s − 1.17·26-s − 2.69·27-s − 0.557·29-s − 3.28·30-s − 0.538·31-s − 1.06·32-s − 3.08·34-s − 2·36-s + 0.640·39-s − 1.42·40-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

\(L(1)\) \(\approx\) \(0.9119827809\)
\(L(\frac12)\) \(\approx\) \(0.9119827809\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 15 T + 122 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 3 T + 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−10.32047630075020485040049107394, −10.30811102765878725475248472929, −9.629752016250532487112023106325, −9.325956418422110678002870577232, −9.153498610713197598951502863832, −8.796033487749241537021023395254, −8.290159621570098065336256311891, −8.042982939212086687303114056667, −7.53737002805107178408428701174, −7.27840674149287705930315240607, −6.15306792176433334455836663977, −6.11154653709719775935433643482, −5.50079942395521476887978407787, −5.15732904004356608295242730285, −3.80894551261859145822128750566, −3.51741954053204866270502564434, −2.73516923260360768465383576781, −2.23886399269075425055084671501, −1.62158487891665727214313765225, −0.70586799034918320325363066594, 0.70586799034918320325363066594, 1.62158487891665727214313765225, 2.23886399269075425055084671501, 2.73516923260360768465383576781, 3.51741954053204866270502564434, 3.80894551261859145822128750566, 5.15732904004356608295242730285, 5.50079942395521476887978407787, 6.11154653709719775935433643482, 6.15306792176433334455836663977, 7.27840674149287705930315240607, 7.53737002805107178408428701174, 8.042982939212086687303114056667, 8.290159621570098065336256311891, 8.796033487749241537021023395254, 9.153498610713197598951502863832, 9.325956418422110678002870577232, 9.629752016250532487112023106325, 10.30811102765878725475248472929, 10.32047630075020485040049107394

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