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Properties

Label 2-370-185.103-c1-0-15
Degree $2$
Conductor $370$
Sign $0.917 + 0.397i$
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  + (0.5 + 0.866i)2-s + (−0.0267 − 0.0997i)3-s + (−0.499 + 0.866i)4-s + (0.206 − 2.22i)5-s + (0.0730 − 0.0730i)6-s + (−0.571 − 2.13i)7-s − 0.999·8-s + (2.58 − 1.49i)9-s + (2.03 − 0.934i)10-s − 2.16i·11-s + (0.0997 + 0.0267i)12-s + (0.466 − 0.807i)13-s + (1.56 − 1.56i)14-s + (−0.227 + 0.0389i)15-s + (−0.5 − 0.866i)16-s + (−4.83 + 2.79i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.0154 − 0.0575i)3-s + (−0.249 + 0.433i)4-s + (0.0923 − 0.995i)5-s + (0.0298 − 0.0298i)6-s + (−0.216 − 0.806i)7-s − 0.353·8-s + (0.862 − 0.498i)9-s + (0.642 − 0.295i)10-s − 0.651i·11-s + (0.0287 + 0.00771i)12-s + (0.129 − 0.224i)13-s + (0.417 − 0.417i)14-s + (−0.0587 + 0.0100i)15-s + (−0.125 − 0.216i)16-s + (−1.17 + 0.677i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.917 + 0.397i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.917 + 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54135 - 0.319219i\)
\(L(\frac12)\) \(\approx\) \(1.54135 - 0.319219i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.206 + 2.22i)T \)
37 \( 1 + (5.95 - 1.24i)T \)
good3 \( 1 + (0.0267 + 0.0997i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.571 + 2.13i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 2.16iT - 11T^{2} \)
13 \( 1 + (-0.466 + 0.807i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.83 - 2.79i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.52 + 1.21i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 9.14T + 23T^{2} \)
29 \( 1 + (3.75 - 3.75i)T - 29iT^{2} \)
31 \( 1 + (-2.61 - 2.61i)T + 31iT^{2} \)
41 \( 1 + (-0.618 - 0.356i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 1.79T + 43T^{2} \)
47 \( 1 + (0.916 - 0.916i)T - 47iT^{2} \)
53 \( 1 + (0.460 - 1.72i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.639 + 2.38i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.94 + 1.86i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (8.38 - 2.24i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (6.02 - 10.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.72 - 7.72i)T - 73iT^{2} \)
79 \( 1 + (-12.3 + 3.31i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-0.483 + 1.80i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (5.35 + 1.43i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−11.43124265337953202661373084318, −10.38343108588054639981891867158, −9.224790014172563531241045719252, −8.616835941885112461579651801676, −7.36499489273964284794532915843, −6.66524186593462999933776914150, −5.41709265704847823345885864773, −4.46304323829225197099040009482, −3.43962055081459098768382966984, −1.07107609629666971458569118276, 2.01371101667645423111732603436, 3.03590936104598189374255278725, 4.40437178494964049851167712324, 5.46613736408377548087205664683, 6.73024376451554234199989065581, 7.46524810083922439389761305172, 9.076469955413178438278271708867, 9.725318122101739277343065834317, 10.68417729806322736375944680634, 11.40628058810532701628118714015

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