Properties

Label 4-7098e2-1.1-c1e2-0-19
Degree $4$
Conductor $50381604$
Sign $1$
Analytic cond. $3212.37$
Root an. cond. $7.52846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s + 3·4-s − 3·5-s + 4·6-s − 2·7-s + 4·8-s + 3·9-s − 6·10-s + 11-s + 6·12-s − 4·14-s − 6·15-s + 5·16-s − 10·17-s + 6·18-s + 3·19-s − 9·20-s − 4·21-s + 2·22-s − 10·23-s + 8·24-s + 25-s + 4·27-s − 6·28-s − 2·29-s − 12·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.34·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 1.89·10-s + 0.301·11-s + 1.73·12-s − 1.06·14-s − 1.54·15-s + 5/4·16-s − 2.42·17-s + 1.41·18-s + 0.688·19-s − 2.01·20-s − 0.872·21-s + 0.426·22-s − 2.08·23-s + 1.63·24-s + 1/5·25-s + 0.769·27-s − 1.13·28-s − 0.371·29-s − 2.19·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50381604\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3212.37\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7098} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 50381604,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T - 32 T^{2} - p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 16 T + 153 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 9 T + 162 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 7 T + 166 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 7 T + 152 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−7.59799023034145298342820721618, −7.52418635389304660684300848863, −6.94035585208253992263829616745, −6.68729223779448540612542105646, −6.34675053330086894152983381453, −6.27606122046596693685449788097, −5.45507343282041274937272789542, −5.30594699752876541552305103223, −4.56163759182246706457856574962, −4.44480970558296690766779435603, −4.03425969554424684416716580851, −3.87875896133642508053166259367, −3.43373743379594914684767598537, −3.14553860213942786847966960567, −2.57719549039600103188700646321, −2.37217099840063851177879909996, −1.72644525597402823508401686277, −1.41161125840142754535699562107, 0, 0, 1.41161125840142754535699562107, 1.72644525597402823508401686277, 2.37217099840063851177879909996, 2.57719549039600103188700646321, 3.14553860213942786847966960567, 3.43373743379594914684767598537, 3.87875896133642508053166259367, 4.03425969554424684416716580851, 4.44480970558296690766779435603, 4.56163759182246706457856574962, 5.30594699752876541552305103223, 5.45507343282041274937272789542, 6.27606122046596693685449788097, 6.34675053330086894152983381453, 6.68729223779448540612542105646, 6.94035585208253992263829616745, 7.52418635389304660684300848863, 7.59799023034145298342820721618

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