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Properties

Label 2-370-185.117-c1-0-4
Degree $2$
Conductor $370$
Sign $0.950 - 0.309i$
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  + 2-s + (−1.41 − 1.41i)3-s + 4-s + (−0.707 + 2.12i)5-s + (−1.41 − 1.41i)6-s + (2.70 + 2.70i)7-s + 8-s + 1.00i·9-s + (−0.707 + 2.12i)10-s + 3.82i·11-s + (−1.41 − 1.41i)12-s + 2.24·13-s + (2.70 + 2.70i)14-s + (4 − 1.99i)15-s + 16-s − 1.82i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.816 − 0.816i)3-s + 0.5·4-s + (−0.316 + 0.948i)5-s + (−0.577 − 0.577i)6-s + (1.02 + 1.02i)7-s + 0.353·8-s + 0.333i·9-s + (−0.223 + 0.670i)10-s + 1.15i·11-s + (−0.408 − 0.408i)12-s + 0.621·13-s + (0.723 + 0.723i)14-s + (1.03 − 0.516i)15-s + 0.250·16-s − 0.443i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.950 - 0.309i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.950 - 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66132 + 0.263331i\)
\(L(\frac12)\) \(\approx\) \(1.66132 + 0.263331i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + (0.707 - 2.12i)T \)
37 \( 1 + (-3.53 + 4.94i)T \)
good3 \( 1 + (1.41 + 1.41i)T + 3iT^{2} \)
7 \( 1 + (-2.70 - 2.70i)T + 7iT^{2} \)
11 \( 1 - 3.82iT - 11T^{2} \)
13 \( 1 - 2.24T + 13T^{2} \)
17 \( 1 + 1.82iT - 17T^{2} \)
19 \( 1 + (-5.82 + 5.82i)T - 19iT^{2} \)
23 \( 1 + 2.58T + 23T^{2} \)
29 \( 1 + (-6.70 - 6.70i)T + 29iT^{2} \)
31 \( 1 + (4.12 - 4.12i)T - 31iT^{2} \)
41 \( 1 - 7iT - 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + (8.24 + 8.24i)T + 47iT^{2} \)
53 \( 1 + (-2.12 + 2.12i)T - 53iT^{2} \)
59 \( 1 + (-1.17 + 1.17i)T - 59iT^{2} \)
61 \( 1 + (-9.29 + 9.29i)T - 61iT^{2} \)
67 \( 1 + (1.24 - 1.24i)T - 67iT^{2} \)
71 \( 1 + 0.343T + 71T^{2} \)
73 \( 1 + (6 + 6i)T + 73iT^{2} \)
79 \( 1 + (9.65 - 9.65i)T - 79iT^{2} \)
83 \( 1 + (0.828 - 0.828i)T - 83iT^{2} \)
89 \( 1 + (7.41 + 7.41i)T + 89iT^{2} \)
97 \( 1 + 7iT - 97T^{2} \)
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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

−11.52789226248763337865737148019, −11.15708581115937146559432046754, −9.811983099384916337338456958785, −8.400428276260270582516337840592, −7.18781543179688739543904439372, −6.78070927117195255755487554750, −5.57024841441011423800103140687, −4.79013137341406697398926023236, −3.09452266044948434671060465352, −1.76072395497728198055369698290, 1.19982657424548340600331391823, 3.74644271416177653164429424836, 4.36227433335859498757473340180, 5.36440753677750694281223768893, 6.03689693147709609272056939317, 7.77757504223283885940406960479, 8.298221075009295443874887382909, 9.876294677962274034059303705238, 10.64920312406546004892775030420, 11.55776847071547376696664654070

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