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Properties

Label 2-371-371.69-c0-0-0
Degree $2$
Conductor $371$
Sign $0.910 - 0.413i$
Analytic cond. $0.185153$
Root an. cond. $0.430294$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.45 + 0.358i)2-s + (1.09 + 0.576i)4-s + (−0.970 − 0.239i)7-s + (0.270 + 0.239i)8-s + (−0.354 + 0.935i)9-s + (−0.0854 − 0.704i)11-s + (−1.32 − 0.695i)14-s + (−0.398 − 0.576i)16-s + (−0.850 + 1.23i)18-s + (0.127 − 1.05i)22-s + 0.241·23-s + (−0.748 − 0.663i)25-s + (−0.928 − 0.822i)28-s + (−0.180 + 1.48i)29-s + (−0.499 − 1.31i)32-s + ⋯
L(s)  = 1  + (1.45 + 0.358i)2-s + (1.09 + 0.576i)4-s + (−0.970 − 0.239i)7-s + (0.270 + 0.239i)8-s + (−0.354 + 0.935i)9-s + (−0.0854 − 0.704i)11-s + (−1.32 − 0.695i)14-s + (−0.398 − 0.576i)16-s + (−0.850 + 1.23i)18-s + (0.127 − 1.05i)22-s + 0.241·23-s + (−0.748 − 0.663i)25-s + (−0.928 − 0.822i)28-s + (−0.180 + 1.48i)29-s + (−0.499 − 1.31i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(371\)    =    \(7 \cdot 53\)
Sign: $0.910 - 0.413i$
Analytic conductor: \(0.185153\)
Root analytic conductor: \(0.430294\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{371} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 371,\ (\ :0),\ 0.910 - 0.413i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.458044378\)
\(L(\frac12)\) \(\approx\) \(1.458044378\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.970 + 0.239i)T \)
53 \( 1 + (0.354 + 0.935i)T \)
good2 \( 1 + (-1.45 - 0.358i)T + (0.885 + 0.464i)T^{2} \)
3 \( 1 + (0.354 - 0.935i)T^{2} \)
5 \( 1 + (0.748 + 0.663i)T^{2} \)
11 \( 1 + (0.0854 + 0.704i)T + (-0.970 + 0.239i)T^{2} \)
13 \( 1 + (-0.568 + 0.822i)T^{2} \)
17 \( 1 + (-0.120 + 0.992i)T^{2} \)
19 \( 1 + (-0.568 + 0.822i)T^{2} \)
23 \( 1 - 0.241T + T^{2} \)
29 \( 1 + (0.180 - 1.48i)T + (-0.970 - 0.239i)T^{2} \)
31 \( 1 + (0.970 + 0.239i)T^{2} \)
37 \( 1 + (-1.00 - 1.45i)T + (-0.354 + 0.935i)T^{2} \)
41 \( 1 + (0.970 - 0.239i)T^{2} \)
43 \( 1 + (-0.645 + 0.935i)T + (-0.354 - 0.935i)T^{2} \)
47 \( 1 + (0.748 - 0.663i)T^{2} \)
59 \( 1 + (0.748 - 0.663i)T^{2} \)
61 \( 1 + (-0.120 - 0.992i)T^{2} \)
67 \( 1 + (0.627 + 0.329i)T + (0.568 + 0.822i)T^{2} \)
71 \( 1 + (1.10 - 1.59i)T + (-0.354 - 0.935i)T^{2} \)
73 \( 1 + (-0.120 + 0.992i)T^{2} \)
79 \( 1 + (-1.88 + 0.464i)T + (0.885 - 0.464i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.120 + 0.992i)T^{2} \)
97 \( 1 + (0.748 + 0.663i)T^{2} \)
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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

−11.93004405439437154341000030131, −11.00306583560990501523202793830, −10.01748994927772462230262288042, −8.817520319977633752869458891037, −7.62191633608491041546536753998, −6.58553872749724953879454483290, −5.78723944268367196396568189586, −4.85490713659582635705957109625, −3.66486409950945913982218231049, −2.71470648090252069791242726399, 2.44290808503215500511718442258, 3.50287149981248467922351456136, 4.40265194730839423614365635112, 5.77573701306554997207741831691, 6.28286027366962878237377888344, 7.51142382212087559800531612510, 9.079848636845965552648519787560, 9.731170533617016811939494016631, 11.02200457966535831048418001990, 11.88967471406745043214072225437

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