Properties

Label 2-1150-1.1-c3-0-65
Degree $2$
Conductor $1150$
Sign $-1$
Analytic cond. $67.8521$
Root an. cond. $8.23724$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 4·4-s + 2·6-s + 18·7-s − 8·8-s − 26·9-s − 32·11-s − 4·12-s + 47·13-s − 36·14-s + 16·16-s − 20·17-s + 52·18-s + 36·19-s − 18·21-s + 64·22-s + 23·23-s + 8·24-s − 94·26-s + 53·27-s + 72·28-s − 27·29-s − 33·31-s − 32·32-s + 32·33-s + 40·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.192·3-s + 1/2·4-s + 0.136·6-s + 0.971·7-s − 0.353·8-s − 0.962·9-s − 0.877·11-s − 0.0962·12-s + 1.00·13-s − 0.687·14-s + 1/4·16-s − 0.285·17-s + 0.680·18-s + 0.434·19-s − 0.187·21-s + 0.620·22-s + 0.208·23-s + 0.0680·24-s − 0.709·26-s + 0.377·27-s + 0.485·28-s − 0.172·29-s − 0.191·31-s − 0.176·32-s + 0.168·33-s + 0.201·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(67.8521\)
Root analytic conductor: \(8.23724\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1150} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1150,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
23 \( 1 - p T \)
good3 \( 1 + T + p^{3} T^{2} \)
7 \( 1 - 18 T + p^{3} T^{2} \)
11 \( 1 + 32 T + p^{3} T^{2} \)
13 \( 1 - 47 T + p^{3} T^{2} \)
17 \( 1 + 20 T + p^{3} T^{2} \)
19 \( 1 - 36 T + p^{3} T^{2} \)
29 \( 1 + 27 T + p^{3} T^{2} \)
31 \( 1 + 33 T + p^{3} T^{2} \)
37 \( 1 + 56 T + p^{3} T^{2} \)
41 \( 1 + 157 T + p^{3} T^{2} \)
43 \( 1 + 18 T + p^{3} T^{2} \)
47 \( 1 + 65 T + p^{3} T^{2} \)
53 \( 1 - 14 T + p^{3} T^{2} \)
59 \( 1 + 744 T + p^{3} T^{2} \)
61 \( 1 - 552 T + p^{3} T^{2} \)
67 \( 1 - 156 T + p^{3} T^{2} \)
71 \( 1 - 699 T + p^{3} T^{2} \)
73 \( 1 - 609 T + p^{3} T^{2} \)
79 \( 1 + 644 T + p^{3} T^{2} \)
83 \( 1 + 512 T + p^{3} T^{2} \)
89 \( 1 + 102 T + p^{3} T^{2} \)
97 \( 1 + 578 T + p^{3} T^{2} \)
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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

−8.801694185185529980426259609488, −8.298282708418698250074903771455, −7.62030510933175435269731195022, −6.54957855607424123816460037942, −5.62161088786731270119855679104, −4.91806521365120331237022310974, −3.50425058643302021168851113884, −2.41704504641600854839838949231, −1.27910741156678225549380618519, 0, 1.27910741156678225549380618519, 2.41704504641600854839838949231, 3.50425058643302021168851113884, 4.91806521365120331237022310974, 5.62161088786731270119855679104, 6.54957855607424123816460037942, 7.62030510933175435269731195022, 8.298282708418698250074903771455, 8.801694185185529980426259609488

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