Properties

Label 4-507e2-1.1-c1e2-0-24
Degree $4$
Conductor $257049$
Sign $1$
Analytic cond. $16.3896$
Root an. cond. $2.01206$
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 + 3-s + 2·4-s + 2·5-s + 6-s + 2·7-s + 5·8-s + 2·10-s − 2·11-s + 2·12-s + 2·14-s + 2·15-s + 5·16-s + 7·17-s − 6·19-s + 4·20-s + 2·21-s − 2·22-s + 6·23-s + 5·24-s − 7·25-s − 27-s + 4·28-s + 29-s + 2·30-s − 8·31-s + 10·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 4-s + 0.894·5-s + 0.408·6-s + 0.755·7-s + 1.76·8-s + 0.632·10-s − 0.603·11-s + 0.577·12-s + 0.534·14-s + 0.516·15-s + 5/4·16-s + 1.69·17-s − 1.37·19-s + 0.894·20-s + 0.436·21-s − 0.426·22-s + 1.25·23-s + 1.02·24-s − 7/5·25-s − 0.192·27-s + 0.755·28-s + 0.185·29-s + 0.365·30-s − 1.43·31-s + 1.76·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(257049\)    =    \(3^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(16.3896\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{507} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 257049,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.280970632\)
\(L(\frac12)\) \(\approx\) \(5.280970632\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
13 \( 1 \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−10.94436107747086797608162783017, −10.90087348568912667803778320753, −10.19429523630352836345527986408, −10.02273523098508057396530509239, −9.434663920059725660171715396629, −8.933070099298467142499763039899, −8.179248265142104393537326013376, −7.984221796721787185930510575880, −7.43575146998605950160025344689, −7.26892067999449572371863892402, −6.23994384736637984784911652788, −6.18405259507606464087169152397, −5.39297393756147015671675905856, −5.05385001149627792632412189378, −4.52355432452460168761623664435, −3.84653001130115399933245339761, −3.19414375359183809376105615785, −2.55000855242731765819362805204, −1.81955210401952935995987153766, −1.52782832581302801348174105880, 1.52782832581302801348174105880, 1.81955210401952935995987153766, 2.55000855242731765819362805204, 3.19414375359183809376105615785, 3.84653001130115399933245339761, 4.52355432452460168761623664435, 5.05385001149627792632412189378, 5.39297393756147015671675905856, 6.18405259507606464087169152397, 6.23994384736637984784911652788, 7.26892067999449572371863892402, 7.43575146998605950160025344689, 7.984221796721787185930510575880, 8.179248265142104393537326013376, 8.933070099298467142499763039899, 9.434663920059725660171715396629, 10.02273523098508057396530509239, 10.19429523630352836345527986408, 10.90087348568912667803778320753, 10.94436107747086797608162783017

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