Properties

Label 2-370-1.1-c3-0-34
Degree $2$
Conductor $370$
Sign $-1$
Analytic cond. $21.8307$
Root an. cond. $4.67233$
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 + 5·5-s + 2·6-s − 25·7-s + 8·8-s − 26·9-s + 10·10-s + 9·11-s + 4·12-s − 76·13-s − 50·14-s + 5·15-s + 16·16-s − 24·17-s − 52·18-s − 40·19-s + 20·20-s − 25·21-s + 18·22-s − 72·23-s + 8·24-s + 25·25-s − 152·26-s − 53·27-s − 100·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.192·3-s + 1/2·4-s + 0.447·5-s + 0.136·6-s − 1.34·7-s + 0.353·8-s − 0.962·9-s + 0.316·10-s + 0.246·11-s + 0.0962·12-s − 1.62·13-s − 0.954·14-s + 0.0860·15-s + 1/4·16-s − 0.342·17-s − 0.680·18-s − 0.482·19-s + 0.223·20-s − 0.259·21-s + 0.174·22-s − 0.652·23-s + 0.0680·24-s + 1/5·25-s − 1.14·26-s − 0.377·27-s − 0.674·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-1$
Analytic conductor: \(21.8307\)
Root analytic conductor: \(4.67233\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{370} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 370,\ (\ :3/2),\ -1)\)

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 \( 1 - p T \)
37 \( 1 - p T \)
good3 \( 1 - T + p^{3} T^{2} \)
7 \( 1 + 25 T + p^{3} T^{2} \)
11 \( 1 - 9 T + p^{3} T^{2} \)
13 \( 1 + 76 T + p^{3} T^{2} \)
17 \( 1 + 24 T + p^{3} T^{2} \)
19 \( 1 + 40 T + p^{3} T^{2} \)
23 \( 1 + 72 T + p^{3} T^{2} \)
29 \( 1 - 60 T + p^{3} T^{2} \)
31 \( 1 - 26 T + p^{3} T^{2} \)
41 \( 1 - 267 T + p^{3} T^{2} \)
43 \( 1 + 382 T + p^{3} T^{2} \)
47 \( 1 - 267 T + p^{3} T^{2} \)
53 \( 1 - 171 T + p^{3} T^{2} \)
59 \( 1 - 396 T + p^{3} T^{2} \)
61 \( 1 + 898 T + p^{3} T^{2} \)
67 \( 1 + 676 T + p^{3} T^{2} \)
71 \( 1 + 21 T + p^{3} T^{2} \)
73 \( 1 + 691 T + p^{3} T^{2} \)
79 \( 1 + 394 T + p^{3} T^{2} \)
83 \( 1 - 309 T + p^{3} T^{2} \)
89 \( 1 + 918 T + p^{3} T^{2} \)
97 \( 1 + 766 T + p^{3} T^{2} \)
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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

−10.42617987181888158078801336018, −9.694104063659642656775185664330, −8.788118695289241191045997287123, −7.44251464770785429034068769221, −6.45494147639985352612688981137, −5.72201684010363084374377095803, −4.48489158449784748873172943482, −3.13769075184624206104279313722, −2.32975337272063355173174635396, 0, 2.32975337272063355173174635396, 3.13769075184624206104279313722, 4.48489158449784748873172943482, 5.72201684010363084374377095803, 6.45494147639985352612688981137, 7.44251464770785429034068769221, 8.788118695289241191045997287123, 9.694104063659642656775185664330, 10.42617987181888158078801336018

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