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Properties

Label 2-370-185.84-c1-0-6
Degree $2$
Conductor $370$
Sign $0.978 + 0.206i$
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  + (−0.866 − 0.5i)2-s + (−2.18 + 1.26i)3-s + (0.499 + 0.866i)4-s + (−1.37 − 1.76i)5-s + 2.52·6-s + (−1.91 + 1.10i)7-s − 0.999i·8-s + (1.68 − 2.92i)9-s + (0.311 + 2.21i)10-s − 2.68·11-s + (−2.18 − 1.26i)12-s + (4.10 − 2.36i)13-s + 2.21·14-s + (5.23 + 2.11i)15-s + (−0.5 + 0.866i)16-s + (0.596 + 0.344i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−1.26 + 0.729i)3-s + (0.249 + 0.433i)4-s + (−0.615 − 0.788i)5-s + 1.03·6-s + (−0.724 + 0.418i)7-s − 0.353i·8-s + (0.562 − 0.975i)9-s + (0.0983 + 0.700i)10-s − 0.810·11-s + (−0.631 − 0.364i)12-s + (1.13 − 0.657i)13-s + 0.591·14-s + (1.35 + 0.546i)15-s + (−0.125 + 0.216i)16-s + (0.144 + 0.0835i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.978 + 0.206i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.978 + 0.206i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.479135 - 0.0499037i\)
\(L(\frac12)\) \(\approx\) \(0.479135 - 0.0499037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (1.37 + 1.76i)T \)
37 \( 1 + (-6.08 + 0.178i)T \)
good3 \( 1 + (2.18 - 1.26i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.91 - 1.10i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.68T + 11T^{2} \)
13 \( 1 + (-4.10 + 2.36i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.596 - 0.344i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.59 - 4.48i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.21iT - 23T^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
31 \( 1 - 6.73T + 31T^{2} \)
41 \( 1 + (3.71 + 6.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 5.90iT - 43T^{2} \)
47 \( 1 - 4.83iT - 47T^{2} \)
53 \( 1 + (0.680 + 0.392i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.130 + 0.225i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.73 + 4.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.31 + 1.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.90 + 8.49i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.1iT - 73T^{2} \)
79 \( 1 + (1.38 + 2.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.67 - 4.42i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.334 - 0.578i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.9iT - 97T^{2} \)
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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

−11.26024453713855563335367141728, −10.41489992789771228079178003145, −9.857149486745898748638374307444, −8.634111906442421512111937721725, −7.928861165218444727081772759895, −6.35052584689315193075738148012, −5.53165498915271438707245228067, −4.43575897724106031829191708697, −3.18928580606314077312060273744, −0.73361927748474593218282558963, 0.840040015590459939474099016018, 3.00199479530917201396311529386, 4.73363910341544187555556054904, 6.17864699271712539159183446598, 6.60838908081440717316307206943, 7.41639383250381823116919916010, 8.390801623091747291562045416827, 9.837792298313751921355865867069, 10.63217822612903731742687520248, 11.40841834149251984307901976904

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