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Properties

Label 2-370-37.11-c1-0-9
Degree $2$
Conductor $370$
Sign $-0.0389 + 0.999i$
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  + (0.866 − 0.5i)2-s + (1.40 − 2.43i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s − 2.81i·6-s + (−0.712 + 1.23i)7-s − 0.999i·8-s + (−2.45 − 4.25i)9-s + 0.999·10-s + 0.135·11-s + (−1.40 − 2.43i)12-s + (2.09 + 1.21i)13-s + 1.42i·14-s + (2.43 − 1.40i)15-s + (−0.5 − 0.866i)16-s + (−2.46 + 1.42i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.812 − 1.40i)3-s + (0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s − 1.14i·6-s + (−0.269 + 0.466i)7-s − 0.353i·8-s + (−0.819 − 1.41i)9-s + 0.316·10-s + 0.0409·11-s + (−0.406 − 0.703i)12-s + (0.582 + 0.336i)13-s + 0.380i·14-s + (0.629 − 0.363i)15-s + (−0.125 − 0.216i)16-s + (−0.597 + 0.345i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0389 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0389 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.0389 + 0.999i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.0389 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68241 - 1.74921i\)
\(L(\frac12)\) \(\approx\) \(1.68241 - 1.74921i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (5.97 - 1.15i)T \)
good3 \( 1 + (-1.40 + 2.43i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.712 - 1.23i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.135T + 11T^{2} \)
13 \( 1 + (-2.09 - 1.21i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.46 - 1.42i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.07 + 1.19i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.45iT - 23T^{2} \)
29 \( 1 - 2.74iT - 29T^{2} \)
31 \( 1 - 4.04iT - 31T^{2} \)
41 \( 1 + (0.490 - 0.849i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 11.8iT - 43T^{2} \)
47 \( 1 - 0.163T + 47T^{2} \)
53 \( 1 + (-0.462 - 0.800i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-12.5 + 7.22i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.11 - 4.68i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.94 + 6.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.98 - 5.16i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.17T + 73T^{2} \)
79 \( 1 + (-4.63 - 2.67i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.00 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.45 - 4.88i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.5iT - 97T^{2} \)
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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

−11.45434123194410520450536997756, −10.38011919093860778045439537157, −9.074182992677133001008236870044, −8.466945340877330603068668304034, −7.09335388550863005219038156815, −6.53984102986900188891591374740, −5.46173618501411458053837746057, −3.71073901819346167564242196344, −2.56197449901494819229598183577, −1.61075566230342158979085169191, 2.57552932383215724432808871871, 3.77715193512332737170182472483, 4.47760779459596197849770255829, 5.58382044216220235401995319245, 6.75324267673306916899418508402, 8.145310006139338329684047515931, 8.846450454867225840982650442771, 9.834909940542927389707170716296, 10.52505490100297969114108367657, 11.48052843205372702161645785097

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