Properties

Label 2-350-1.1-c3-0-6
Degree $2$
Conductor $350$
Sign $1$
Analytic cond. $20.6506$
Root an. cond. $4.54430$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 4·4-s − 4·6-s − 7·7-s − 8·8-s − 23·9-s − 27·11-s + 8·12-s + 64·13-s + 14·14-s + 16·16-s + 24·17-s + 46·18-s + 62·19-s − 14·21-s + 54·22-s + 105·23-s − 16·24-s − 128·26-s − 100·27-s − 28·28-s + 141·29-s − 124·31-s − 32·32-s − 54·33-s − 48·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.384·3-s + 1/2·4-s − 0.272·6-s − 0.377·7-s − 0.353·8-s − 0.851·9-s − 0.740·11-s + 0.192·12-s + 1.36·13-s + 0.267·14-s + 1/4·16-s + 0.342·17-s + 0.602·18-s + 0.748·19-s − 0.145·21-s + 0.523·22-s + 0.951·23-s − 0.136·24-s − 0.965·26-s − 0.712·27-s − 0.188·28-s + 0.902·29-s − 0.718·31-s − 0.176·32-s − 0.284·33-s − 0.242·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.367097560\)
\(L(\frac12)\) \(\approx\) \(1.367097560\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
5 \( 1 \)
7 \( 1 + p T \)
good3 \( 1 - 2 T + p^{3} T^{2} \)
11 \( 1 + 27 T + p^{3} T^{2} \)
13 \( 1 - 64 T + p^{3} T^{2} \)
17 \( 1 - 24 T + p^{3} T^{2} \)
19 \( 1 - 62 T + p^{3} T^{2} \)
23 \( 1 - 105 T + p^{3} T^{2} \)
29 \( 1 - 141 T + p^{3} T^{2} \)
31 \( 1 + 4 p T + p^{3} T^{2} \)
37 \( 1 - 439 T + p^{3} T^{2} \)
41 \( 1 + 354 T + p^{3} T^{2} \)
43 \( 1 - 211 T + p^{3} T^{2} \)
47 \( 1 - 102 T + p^{3} T^{2} \)
53 \( 1 - 306 T + p^{3} T^{2} \)
59 \( 1 - 348 T + p^{3} T^{2} \)
61 \( 1 - 410 T + p^{3} T^{2} \)
67 \( 1 - 349 T + p^{3} T^{2} \)
71 \( 1 + 339 T + p^{3} T^{2} \)
73 \( 1 - 70 T + p^{3} T^{2} \)
79 \( 1 - 731 T + p^{3} T^{2} \)
83 \( 1 + 528 T + p^{3} T^{2} \)
89 \( 1 - 960 T + p^{3} T^{2} \)
97 \( 1 + 1340 T + p^{3} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.99337076074995362694527464545, −10.05737658074226236431093792009, −9.064981121128506360375738848594, −8.393428283614674170225810767068, −7.50805508863986077707979634556, −6.30800414976186257160673741172, −5.35998193112450924891206231201, −3.54971603366328936255168652454, −2.59161560455248369418694824516, −0.863195869564064719933081326343, 0.863195869564064719933081326343, 2.59161560455248369418694824516, 3.54971603366328936255168652454, 5.35998193112450924891206231201, 6.30800414976186257160673741172, 7.50805508863986077707979634556, 8.393428283614674170225810767068, 9.064981121128506360375738848594, 10.05737658074226236431093792009, 10.99337076074995362694527464545

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