Properties

Label 2-2352-1.1-c3-0-74
Degree $2$
Conductor $2352$
Sign $-1$
Analytic cond. $138.772$
Root an. cond. $11.7801$
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  − 3·3-s − 3·5-s + 9·9-s + 15·11-s − 64·13-s + 9·15-s + 84·17-s + 16·19-s + 84·23-s − 116·25-s − 27·27-s − 297·29-s + 253·31-s − 45·33-s − 316·37-s + 192·39-s + 360·41-s − 26·43-s − 27·45-s + 30·47-s − 252·51-s + 363·53-s − 45·55-s − 48·57-s + 15·59-s − 118·61-s + 192·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.268·5-s + 1/3·9-s + 0.411·11-s − 1.36·13-s + 0.154·15-s + 1.19·17-s + 0.193·19-s + 0.761·23-s − 0.927·25-s − 0.192·27-s − 1.90·29-s + 1.46·31-s − 0.237·33-s − 1.40·37-s + 0.788·39-s + 1.37·41-s − 0.0922·43-s − 0.0894·45-s + 0.0931·47-s − 0.691·51-s + 0.940·53-s − 0.110·55-s − 0.111·57-s + 0.0330·59-s − 0.247·61-s + 0.366·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(138.772\)
Root analytic conductor: \(11.7801\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2352} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ -1)\)

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 \( 1 \)
3 \( 1 + p T \)
7 \( 1 \)
good5 \( 1 + 3 T + p^{3} T^{2} \)
11 \( 1 - 15 T + p^{3} T^{2} \)
13 \( 1 + 64 T + p^{3} T^{2} \)
17 \( 1 - 84 T + p^{3} T^{2} \)
19 \( 1 - 16 T + p^{3} T^{2} \)
23 \( 1 - 84 T + p^{3} T^{2} \)
29 \( 1 + 297 T + p^{3} T^{2} \)
31 \( 1 - 253 T + p^{3} T^{2} \)
37 \( 1 + 316 T + p^{3} T^{2} \)
41 \( 1 - 360 T + p^{3} T^{2} \)
43 \( 1 + 26 T + p^{3} T^{2} \)
47 \( 1 - 30 T + p^{3} T^{2} \)
53 \( 1 - 363 T + p^{3} T^{2} \)
59 \( 1 - 15 T + p^{3} T^{2} \)
61 \( 1 + 118 T + p^{3} T^{2} \)
67 \( 1 - 370 T + p^{3} T^{2} \)
71 \( 1 - 342 T + p^{3} T^{2} \)
73 \( 1 - 362 T + p^{3} T^{2} \)
79 \( 1 + 467 T + p^{3} T^{2} \)
83 \( 1 + 477 T + p^{3} T^{2} \)
89 \( 1 - 906 T + p^{3} T^{2} \)
97 \( 1 - 503 T + p^{3} T^{2} \)
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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

−8.051341822197891256544931767639, −7.43311285159875573101705360016, −6.81244476913271844781406813867, −5.73611260553641627061674196077, −5.20846763709180696848645789021, −4.25962410511594176365862020256, −3.39663671383686353475577391994, −2.26018664937550813509490664793, −1.07892985303383290882430112999, 0, 1.07892985303383290882430112999, 2.26018664937550813509490664793, 3.39663671383686353475577391994, 4.25962410511594176365862020256, 5.20846763709180696848645789021, 5.73611260553641627061674196077, 6.81244476913271844781406813867, 7.43311285159875573101705360016, 8.051341822197891256544931767639

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