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Properties

Label 2-370-185.82-c1-0-2
Degree $2$
Conductor $370$
Sign $0.142 - 0.989i$
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 + (−1.08 + 0.290i)3-s + (−0.499 + 0.866i)4-s + (1.94 + 1.10i)5-s + (0.793 + 0.793i)6-s + (−1.13 + 0.304i)7-s + 0.999·8-s + (−1.50 + 0.869i)9-s + (−0.0180 − 2.23i)10-s − 1.21i·11-s + (0.290 − 1.08i)12-s + (−1.27 + 2.21i)13-s + (0.830 + 0.830i)14-s + (−2.42 − 0.630i)15-s + (−0.5 − 0.866i)16-s + (−5.18 + 2.99i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.626 + 0.167i)3-s + (−0.249 + 0.433i)4-s + (0.870 + 0.492i)5-s + (0.324 + 0.324i)6-s + (−0.429 + 0.114i)7-s + 0.353·8-s + (−0.502 + 0.289i)9-s + (−0.00571 − 0.707i)10-s − 0.367i·11-s + (0.0838 − 0.313i)12-s + (−0.354 + 0.613i)13-s + (0.222 + 0.222i)14-s + (−0.627 − 0.162i)15-s + (−0.125 − 0.216i)16-s + (−1.25 + 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.142 - 0.989i$
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: \((2,\ 370,\ (\ :1/2),\ 0.142 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.500695 + 0.433715i\)
\(L(\frac12)\) \(\approx\) \(0.500695 + 0.433715i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 - 1.10i)T \)
37 \( 1 + (2.24 - 5.65i)T \)
good3 \( 1 + (1.08 - 0.290i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.13 - 0.304i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + 1.21iT - 11T^{2} \)
13 \( 1 + (1.27 - 2.21i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.18 - 2.99i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.48 - 5.55i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 2.78T + 23T^{2} \)
29 \( 1 + (-3.89 - 3.89i)T + 29iT^{2} \)
31 \( 1 + (-1.69 + 1.69i)T - 31iT^{2} \)
41 \( 1 + (-8.25 - 4.76i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 5.25T + 43T^{2} \)
47 \( 1 + (3.72 + 3.72i)T + 47iT^{2} \)
53 \( 1 + (-5.06 - 1.35i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.984 - 0.263i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.684 + 2.55i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (3.05 + 11.4i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.54 - 4.41i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.93 + 5.93i)T + 73iT^{2} \)
79 \( 1 + (3.10 + 11.5i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-16.7 - 4.49i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-1.51 + 5.66i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 - 8.11iT - 97T^{2} \)
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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

−11.45621910747378556980532970950, −10.61615438210657334740615238199, −10.02621101070039354018711039260, −9.070682680507857119754117944361, −8.089210874172518721617330522546, −6.58907387836515967861353605804, −5.94869721916902414922274541017, −4.67574766170663934794542415287, −3.17367204361978084956086257292, −1.92136472129414354862226981891, 0.52210568820992484450717468089, 2.54860200622626845836229713539, 4.63417321543780852833415720597, 5.50587846585797409587885006101, 6.40970719929630410528427001985, 7.15513568975347813501372167979, 8.558004051188426362569493460216, 9.298394965730431556284637136929, 10.08557484645488048145025359546, 11.10683616665311208564961738898

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