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Properties

Label 2-370-37.12-c1-0-2
Degree $2$
Conductor $370$
Sign $-0.129 - 0.991i$
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  + (−0.766 + 0.642i)2-s + (0.992 + 0.832i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)5-s − 1.29·6-s + (−3.09 + 1.12i)7-s + (0.500 + 0.866i)8-s + (−0.229 − 1.30i)9-s + (−0.5 + 0.866i)10-s + (3.04 + 5.28i)11-s + (0.992 − 0.832i)12-s + (−0.733 + 4.15i)13-s + (1.64 − 2.85i)14-s + (1.21 + 0.443i)15-s + (−0.939 − 0.342i)16-s + (0.443 + 2.51i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (0.572 + 0.480i)3-s + (0.0868 − 0.492i)4-s + (0.420 − 0.152i)5-s − 0.528·6-s + (−1.17 + 0.426i)7-s + (0.176 + 0.306i)8-s + (−0.0764 − 0.433i)9-s + (−0.158 + 0.273i)10-s + (0.919 + 1.59i)11-s + (0.286 − 0.240i)12-s + (−0.203 + 1.15i)13-s + (0.440 − 0.763i)14-s + (0.314 + 0.114i)15-s + (−0.234 − 0.0855i)16-s + (0.107 + 0.609i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.129 - 0.991i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.129 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773667 + 0.881523i\)
\(L(\frac12)\) \(\approx\) \(0.773667 + 0.881523i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (5.66 - 2.20i)T \)
good3 \( 1 + (-0.992 - 0.832i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (3.09 - 1.12i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-3.04 - 5.28i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.733 - 4.15i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.443 - 2.51i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (-4.21 - 3.53i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (-0.278 + 0.482i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.76 + 6.52i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
41 \( 1 + (1.13 - 6.42i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + 8.25T + 43T^{2} \)
47 \( 1 + (-3.46 + 6.00i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.81 + 0.659i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (9.75 + 3.55i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.19 + 12.4i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-8.69 + 3.16i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (10.3 + 8.72i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 + (-12.4 + 4.51i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-1.17 - 6.64i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (0.306 + 0.111i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-3.19 + 5.53i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.85613489527185386266167189402, −10.01267599182577245488257676912, −9.668365379209894606354896003509, −9.223803367812780731862257064690, −8.086965633150003052108523695168, −6.69262676188924095354222768841, −6.29752268026272405370902842655, −4.69408568116600375109421191189, −3.50962358690131535871463129068, −1.90657681645787510358882065319, 0.942251281592240141920859257593, 2.86348689130910383929093663075, 3.36435438258621595237634664151, 5.39738758119179357363489319448, 6.63305644616305325416162402656, 7.45369478800065966216018655542, 8.556441201879930330749736452321, 9.251254550923781211072093435911, 10.21409599983343478580960257835, 10.98168124186787811273050163536

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