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Properties

Label 2-370-185.103-c1-0-2
Degree $2$
Conductor $370$
Sign $-0.980 - 0.195i$
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 + (0.686 + 2.56i)3-s + (−0.499 + 0.866i)4-s + (−1.84 − 1.25i)5-s + (−1.87 + 1.87i)6-s + (0.524 + 1.95i)7-s − 0.999·8-s + (−3.50 + 2.02i)9-s + (0.166 − 2.22i)10-s − 1.12i·11-s + (−2.56 − 0.686i)12-s + (−2.37 + 4.12i)13-s + (−1.43 + 1.43i)14-s + (1.95 − 5.60i)15-s + (−0.5 − 0.866i)16-s + (0.528 − 0.304i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.396 + 1.47i)3-s + (−0.249 + 0.433i)4-s + (−0.826 − 0.562i)5-s + (−0.766 + 0.766i)6-s + (0.198 + 0.739i)7-s − 0.353·8-s + (−1.16 + 0.673i)9-s + (0.0525 − 0.705i)10-s − 0.338i·11-s + (−0.739 − 0.198i)12-s + (−0.659 + 1.14i)13-s + (−0.382 + 0.382i)14-s + (0.505 − 1.44i)15-s + (−0.125 − 0.216i)16-s + (0.128 − 0.0739i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.980 - 0.195i$
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: \((2,\ 370,\ (\ :1/2),\ -0.980 - 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.137403 + 1.39489i\)
\(L(\frac12)\) \(\approx\) \(0.137403 + 1.39489i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.84 + 1.25i)T \)
37 \( 1 + (-4.85 - 3.67i)T \)
good3 \( 1 + (-0.686 - 2.56i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.524 - 1.95i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 1.12iT - 11T^{2} \)
13 \( 1 + (2.37 - 4.12i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.528 + 0.304i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.968 - 0.259i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 5.88T + 23T^{2} \)
29 \( 1 + (-2.37 + 2.37i)T - 29iT^{2} \)
31 \( 1 + (3.83 + 3.83i)T + 31iT^{2} \)
41 \( 1 + (-7.57 - 4.37i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 2.52T + 43T^{2} \)
47 \( 1 + (-4.16 + 4.16i)T - 47iT^{2} \)
53 \( 1 + (0.917 - 3.42i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.81 - 6.77i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (13.4 - 3.60i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-14.0 + 3.75i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.54 - 6.14i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.20 - 8.20i)T - 73iT^{2} \)
79 \( 1 + (-4.51 + 1.20i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (0.924 - 3.44i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-13.2 - 3.55i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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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

−11.73833110397170268186875617161, −11.03327034124782536935211163382, −9.636618304790533874123535940968, −9.020276561213244277707391773625, −8.349938394727327644285043874557, −7.21332587223060126024373832531, −5.71693792823982842736413682515, −4.69815102750783855626899536155, −4.16979849109866009857392240442, −2.89673219238015931195200301451, 0.845278320603483256831348238766, 2.46190135682844976209466728264, 3.46357518513701517600991697144, 4.87377061747192889729961867056, 6.39823083924605014672535678483, 7.42551939539688601407153577218, 7.74359479983343381422076537335, 9.011098295134260608430394058677, 10.46820620425621469306572283936, 11.01491110815263696803527392618

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