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Properties

Label 2-370-185.184-c1-0-5
Degree $2$
Conductor $370$
Sign $0.867 - 0.498i$
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 + 1.76i·3-s + 4-s + (−1.62 − 1.53i)5-s − 1.76i·6-s − 1.22i·7-s − 8-s − 0.105·9-s + (1.62 + 1.53i)10-s + 1.87·11-s + 1.76i·12-s + 6.50·13-s + 1.22i·14-s + (2.69 − 2.86i)15-s + 16-s − 0.765·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.01i·3-s + 0.5·4-s + (−0.728 − 0.685i)5-s − 0.719i·6-s − 0.461i·7-s − 0.353·8-s − 0.0350·9-s + (0.515 + 0.484i)10-s + 0.564·11-s + 0.508i·12-s + 1.80·13-s + 0.326i·14-s + (0.697 − 0.741i)15-s + 0.250·16-s − 0.185·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.867 - 0.498i$
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: \((2,\ 370,\ (\ :1/2),\ 0.867 - 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.941595 + 0.251173i\)
\(L(\frac12)\) \(\approx\) \(0.941595 + 0.251173i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 + 1.53i)T \)
37 \( 1 + (-1.76 + 5.82i)T \)
good3 \( 1 - 1.76iT - 3T^{2} \)
7 \( 1 + 1.22iT - 7T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 - 6.50T + 13T^{2} \)
17 \( 1 + 0.765T + 17T^{2} \)
19 \( 1 - 3.34iT - 19T^{2} \)
23 \( 1 - 1.38T + 23T^{2} \)
29 \( 1 + 1.72iT - 29T^{2} \)
31 \( 1 - 4.11iT - 31T^{2} \)
41 \( 1 - 3.73T + 41T^{2} \)
43 \( 1 - 4.91T + 43T^{2} \)
47 \( 1 + 6.30iT - 47T^{2} \)
53 \( 1 + 2.57iT - 53T^{2} \)
59 \( 1 - 10.5iT - 59T^{2} \)
61 \( 1 + 11.1iT - 61T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 - 0.963T + 71T^{2} \)
73 \( 1 - 9.03iT - 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + 0.00656iT - 83T^{2} \)
89 \( 1 + 4.70iT - 89T^{2} \)
97 \( 1 + 0.403T + 97T^{2} \)
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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

−11.07824747886869337902474607475, −10.65319467326268095857746685454, −9.504174087907260037187396261016, −8.840185474484417401951715016932, −8.031938325251895099260035430879, −6.88535761797039082456678625769, −5.60169504965734155355167032167, −4.17905450228621559789044926836, −3.64250923049939807967731791641, −1.20856738834502712407637389645, 1.15202431365447672837557696624, 2.68347517565279136380534429680, 4.05493135743222481555777632216, 6.07322218884136776963966084042, 6.69890240232247767175370520542, 7.58756830587059274868569191608, 8.420926461727632493373473624607, 9.255633442883291945617117408796, 10.62587184559076903945748741297, 11.32781275735716117572915677398

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