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Properties

Label 2-370-185.184-c1-0-8
Degree $2$
Conductor $370$
Sign $0.968 - 0.250i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.377i·3-s + 4-s + (−1.04 + 1.97i)5-s − 0.377i·6-s − 0.631i·7-s + 8-s + 2.85·9-s + (−1.04 + 1.97i)10-s + 1.24·11-s − 0.377i·12-s + 3.34·13-s − 0.631i·14-s + (0.746 + 0.395i)15-s + 16-s + 3.10·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.218i·3-s + 0.5·4-s + (−0.468 + 0.883i)5-s − 0.154i·6-s − 0.238i·7-s + 0.353·8-s + 0.952·9-s + (−0.331 + 0.624i)10-s + 0.376·11-s − 0.109i·12-s + 0.929·13-s − 0.168i·14-s + (0.192 + 0.102i)15-s + 0.250·16-s + 0.753·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04902 + 0.260611i\)
\(L(\frac12)\) \(\approx\) \(2.04902 + 0.260611i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + (1.04 - 1.97i)T \)
37 \( 1 + (4.10 + 4.48i)T \)
good3 \( 1 + 0.377iT - 3T^{2} \)
7 \( 1 + 0.631iT - 7T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
13 \( 1 - 3.34T + 13T^{2} \)
17 \( 1 - 3.10T + 17T^{2} \)
19 \( 1 - 5.97iT - 19T^{2} \)
23 \( 1 + 7.60T + 23T^{2} \)
29 \( 1 + 9.57iT - 29T^{2} \)
31 \( 1 - 7.26iT - 31T^{2} \)
41 \( 1 + 8.45T + 41T^{2} \)
43 \( 1 + 4.86T + 43T^{2} \)
47 \( 1 + 13.1iT - 47T^{2} \)
53 \( 1 - 7.17iT - 53T^{2} \)
59 \( 1 + 4.36iT - 59T^{2} \)
61 \( 1 + 2.14iT - 61T^{2} \)
67 \( 1 - 11.3iT - 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 - 4.45iT - 73T^{2} \)
79 \( 1 + 8.78iT - 79T^{2} \)
83 \( 1 - 6.63iT - 83T^{2} \)
89 \( 1 + 13.7iT - 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−11.86391605977728200880053630706, −10.33928670801098963741726916937, −10.17550667955299126702811610654, −8.331387648450424906346474109041, −7.50135563478597676652581842105, −6.60959792589456473226612287299, −5.76357120815864462371683480803, −4.08532625308258404663766860404, −3.55946677321607037982958348439, −1.79158704786999929262269630690, 1.49268974262983817099086377144, 3.45715606246204618043320582855, 4.37067580394668454147739842349, 5.27384187098563786199027792902, 6.45270859264557915541434531324, 7.55072336993694859881495665213, 8.573846417888219642703611588560, 9.514461157426103623943952256949, 10.57772688564403866369237988595, 11.62629197279110086425948792138

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