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Properties

Label 2-370-185.64-c1-0-4
Degree $2$
Conductor $370$
Sign $0.0395 - 0.999i$
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 + (−1.67 + 0.967i)3-s + (−0.499 − 0.866i)4-s + (−0.564 − 2.16i)5-s − 1.93i·6-s + (0.358 − 0.207i)7-s + 0.999·8-s + (0.370 − 0.641i)9-s + (2.15 + 0.593i)10-s + 4.06·11-s + (1.67 + 0.967i)12-s + (1.51 + 2.61i)13-s + 0.414i·14-s + (3.03 + 3.07i)15-s + (−0.5 + 0.866i)16-s + (−2.66 + 4.61i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.967 + 0.558i)3-s + (−0.249 − 0.433i)4-s + (−0.252 − 0.967i)5-s − 0.789i·6-s + (0.135 − 0.0782i)7-s + 0.353·8-s + (0.123 − 0.213i)9-s + (0.681 + 0.187i)10-s + 1.22·11-s + (0.483 + 0.279i)12-s + (0.419 + 0.725i)13-s + 0.110i·14-s + (0.784 + 0.794i)15-s + (−0.125 + 0.216i)16-s + (−0.646 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0395 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0395 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.0395 - 0.999i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.0395 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.541409 + 0.520401i\)
\(L(\frac12)\) \(\approx\) \(0.541409 + 0.520401i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.564 + 2.16i)T \)
37 \( 1 + (-6.08 - 0.00221i)T \)
good3 \( 1 + (1.67 - 0.967i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.358 + 0.207i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.06T + 11T^{2} \)
13 \( 1 + (-1.51 - 2.61i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.66 - 4.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.74 - 1.58i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 9.34T + 23T^{2} \)
29 \( 1 - 6.14iT - 29T^{2} \)
31 \( 1 + 4.98iT - 31T^{2} \)
41 \( 1 + (-4.42 - 7.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 2.04T + 43T^{2} \)
47 \( 1 - 11.1iT - 47T^{2} \)
53 \( 1 + (0.0502 + 0.0290i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.89 - 2.25i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.62 + 5.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.76 + 1.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.80 + 3.11i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 12.3iT - 73T^{2} \)
79 \( 1 + (-5.31 + 3.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.46 - 0.847i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.57 + 5.53i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.73T + 97T^{2} \)
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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

−11.22180339305994265216641097148, −10.99792996418191953851335108374, −9.531448406840682333197646411758, −8.948126349733409879379447088746, −8.018680455590947087793262324353, −6.62210280094924893603755265749, −5.92198038335612651033781978940, −4.71284999378331341083549226150, −4.13591620336356180252030908103, −1.28774492409300162859648744156, 0.77789990320389038608549821430, 2.61278669144808503559623052432, 3.91203317917396377319690844736, 5.39340072236835276214481643906, 6.70952716295518654730018397988, 7.04498139081607942724769364199, 8.500335960259171713959459137193, 9.440097954797232137029452903201, 10.63241023127798257901741009786, 11.35224883477555688622222389814

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