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Properties

Label 2-370-185.159-c1-0-2
Degree $2$
Conductor $370$
Sign $0.766 - 0.642i$
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 + (−2.51 − 1.45i)3-s + (−0.499 + 0.866i)4-s + (−2.23 − 0.0919i)5-s − 2.90i·6-s + (0.191 + 0.110i)7-s − 0.999·8-s + (2.73 + 4.73i)9-s + (−1.03 − 1.98i)10-s + 6.08·11-s + (2.51 − 1.45i)12-s + (−0.0723 + 0.125i)13-s + 0.221i·14-s + (5.49 + 3.48i)15-s + (−0.5 − 0.866i)16-s + (1.31 + 2.26i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−1.45 − 0.839i)3-s + (−0.249 + 0.433i)4-s + (−0.999 − 0.0411i)5-s − 1.18i·6-s + (0.0725 + 0.0418i)7-s − 0.353·8-s + (0.910 + 1.57i)9-s + (−0.328 − 0.626i)10-s + 1.83·11-s + (0.727 − 0.419i)12-s + (−0.0200 + 0.0347i)13-s + 0.0591i·14-s + (1.41 + 0.898i)15-s + (−0.125 − 0.216i)16-s + (0.317 + 0.550i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.816751 + 0.296832i\)
\(L(\frac12)\) \(\approx\) \(0.816751 + 0.296832i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.0919i)T \)
37 \( 1 + (-5.49 + 2.61i)T \)
good3 \( 1 + (2.51 + 1.45i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.191 - 0.110i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 6.08T + 11T^{2} \)
13 \( 1 + (0.0723 - 0.125i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.31 - 2.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.28 - 1.89i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.277T + 23T^{2} \)
29 \( 1 + 1.60iT - 29T^{2} \)
31 \( 1 - 0.776iT - 31T^{2} \)
41 \( 1 + (5.07 - 8.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 6.51T + 43T^{2} \)
47 \( 1 - 7.42iT - 47T^{2} \)
53 \( 1 + (3.13 - 1.80i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-11.3 + 6.54i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.82 + 5.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.9 - 7.48i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.50 - 11.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.99iT - 73T^{2} \)
79 \( 1 + (-5.73 - 3.30i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.52 + 2.03i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (10.8 - 6.27i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.20T + 97T^{2} \)
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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

−11.61034726753046386188187684667, −11.17542835951460810065548827128, −9.655113413960986607647268896364, −8.328062303622385807880221085762, −7.42656713016544021981306754427, −6.62978714965017499832753081598, −5.93661529945715949336370523048, −4.75602870844347456506077058595, −3.72652172439632242541086518104, −1.14076848845333570410873036781, 0.852963131243367721629592293188, 3.48934413481140497473723631237, 4.28789680378284723200744567886, 5.13651352753643876193758112397, 6.27896414011245811116519942015, 7.23182694540734613855652475868, 8.920295351646193578762887307188, 9.715472715713850706674141271904, 10.71636626614049708928624808638, 11.52283970832480542987343372008

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