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Properties

Label 2-370-185.117-c1-0-13
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 + (1.29 + 1.29i)7-s + 8-s + 1.00i·9-s + (0.707 − 2.12i)10-s − 1.82i·11-s + (1.41 + 1.41i)12-s − 6.24·13-s + (1.29 + 1.29i)14-s + (4 − 1.99i)15-s + 16-s + 3.82i·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 + (0.488 + 0.488i)7-s + 0.353·8-s + 0.333i·9-s + (0.223 − 0.670i)10-s − 0.551i·11-s + (0.408 + 0.408i)12-s − 1.73·13-s + (0.345 + 0.345i)14-s + (1.03 − 0.516i)15-s + 0.250·16-s + 0.928i·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

\(L(1)\) \(\approx\) \(2.61201 + 0.414021i\)
\(L(\frac12)\) \(\approx\) \(2.61201 + 0.414021i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 - 1.29i)T + 7iT^{2} \)
11 \( 1 + 1.82iT - 11T^{2} \)
13 \( 1 + 6.24T + 13T^{2} \)
17 \( 1 - 3.82iT - 17T^{2} \)
19 \( 1 + (-0.171 + 0.171i)T - 19iT^{2} \)
23 \( 1 + 5.41T + 23T^{2} \)
29 \( 1 + (-5.29 - 5.29i)T + 29iT^{2} \)
31 \( 1 + (-0.121 + 0.121i)T - 31iT^{2} \)
41 \( 1 - 7iT - 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + (-0.242 - 0.242i)T + 47iT^{2} \)
53 \( 1 + (2.12 - 2.12i)T - 53iT^{2} \)
59 \( 1 + (-6.82 + 6.82i)T - 59iT^{2} \)
61 \( 1 + (-10.7 + 10.7i)T - 61iT^{2} \)
67 \( 1 + (-7.24 + 7.24i)T - 67iT^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + (6 + 6i)T + 73iT^{2} \)
79 \( 1 + (-1.65 + 1.65i)T - 79iT^{2} \)
83 \( 1 + (-4.82 + 4.82i)T - 83iT^{2} \)
89 \( 1 + (4.58 + 4.58i)T + 89iT^{2} \)
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

−11.75454918749987171530844110417, −10.28640424309537244783467377120, −9.686823563070493776798197407820, −8.596558859571141880330200442739, −8.048651734215428306899751421560, −6.43211338321903990418147799861, −5.18307520199737626130753350306, −4.58666013020708072058807321483, −3.34956284265200716544459094383, −2.04926673312470587617590228006, 2.08513470684197744533469345544, 2.74363814732258187064109732597, 4.27153199826017436133056020815, 5.47223128330950021218278473162, 7.07044190748736069756951874833, 7.18144812819960075616334839422, 8.197829083070458488490163786106, 9.747286884078717412066433768017, 10.38320691111141527980330636321, 11.64265160033865414175365389346

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