Properties

Label 2-370-185.82-c1-0-10
Degree $2$
Conductor $370$
Sign $0.513 - 0.857i$
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 + (1.46 − 0.392i)3-s + (−0.499 + 0.866i)4-s + (1.30 + 1.81i)5-s + (1.07 + 1.07i)6-s + (2.26 − 0.607i)7-s − 0.999·8-s + (−0.603 + 0.348i)9-s + (−0.917 + 2.03i)10-s − 2.09i·11-s + (−0.392 + 1.46i)12-s + (−0.141 + 0.244i)13-s + (1.65 + 1.65i)14-s + (2.62 + 2.14i)15-s + (−0.5 − 0.866i)16-s + (−3.18 + 1.84i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.846 − 0.226i)3-s + (−0.249 + 0.433i)4-s + (0.584 + 0.811i)5-s + (0.438 + 0.438i)6-s + (0.856 − 0.229i)7-s − 0.353·8-s + (−0.201 + 0.116i)9-s + (−0.290 + 0.644i)10-s − 0.632i·11-s + (−0.113 + 0.423i)12-s + (−0.0391 + 0.0678i)13-s + (0.443 + 0.443i)14-s + (0.678 + 0.554i)15-s + (−0.125 − 0.216i)16-s + (−0.773 + 0.446i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.513 - 0.857i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.513 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94120 + 1.09993i\)
\(L(\frac12)\) \(\approx\) \(1.94120 + 1.09993i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 - 1.81i)T \)
37 \( 1 + (-2.05 + 5.72i)T \)
good3 \( 1 + (-1.46 + 0.392i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-2.26 + 0.607i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + 2.09iT - 11T^{2} \)
13 \( 1 + (0.141 - 0.244i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.18 - 1.84i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.26 + 4.71i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 3.38T + 23T^{2} \)
29 \( 1 + (0.0432 + 0.0432i)T + 29iT^{2} \)
31 \( 1 + (-1.04 + 1.04i)T - 31iT^{2} \)
41 \( 1 + (-2.66 - 1.53i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 4.22T + 43T^{2} \)
47 \( 1 + (4.68 + 4.68i)T + 47iT^{2} \)
53 \( 1 + (5.40 + 1.44i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.53 + 0.678i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.33 + 4.98i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.01 + 7.50i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.71 - 6.43i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.07 + 3.07i)T + 73iT^{2} \)
79 \( 1 + (0.207 + 0.775i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-10.1 - 2.71i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (2.69 - 10.0i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−11.23330362650279833888391475923, −10.93167243805810181775682741294, −9.398530478651781020389601175224, −8.616584453036198287070087589459, −7.76184374529134744419711879654, −6.86956158384006751204828025604, −5.86310646468111693809509501230, −4.67283435399280135675969724822, −3.25681183173581598171582739901, −2.17918328633396207377060323126, 1.63888041004520030648804205775, 2.74426484823695025665729927418, 4.24010720350195798271453006725, 5.04292395315460555106004955068, 6.18648353715977122059280509979, 7.83681251625599061102842027825, 8.719777389069932911102933138287, 9.353269154477287586548755277696, 10.22789927604243227127227505326, 11.36122326063762166288796491795

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