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Properties

Label 2-370-185.142-c1-0-5
Degree $2$
Conductor $370$
Sign $-0.480 - 0.877i$
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  + i·2-s + (1.70 + 1.70i)3-s − 4-s + (−1.41 − 1.73i)5-s + (−1.70 + 1.70i)6-s + (2.82 + 2.82i)7-s − i·8-s + 2.79i·9-s + (1.73 − 1.41i)10-s + 2.94i·11-s + (−1.70 − 1.70i)12-s + 0.738i·13-s + (−2.82 + 2.82i)14-s + (0.542 − 5.35i)15-s + 16-s − 6.61·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.982 + 0.982i)3-s − 0.5·4-s + (−0.632 − 0.774i)5-s + (−0.694 + 0.694i)6-s + (1.06 + 1.06i)7-s − 0.353i·8-s + 0.930i·9-s + (0.547 − 0.447i)10-s + 0.887i·11-s + (−0.491 − 0.491i)12-s + 0.204i·13-s + (−0.755 + 0.755i)14-s + (0.140 − 1.38i)15-s + 0.250·16-s − 1.60·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.480 - 0.877i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.480 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.856168 + 1.44464i\)
\(L(\frac12)\) \(\approx\) \(0.856168 + 1.44464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (1.41 + 1.73i)T \)
37 \( 1 + (-0.209 + 6.07i)T \)
good3 \( 1 + (-1.70 - 1.70i)T + 3iT^{2} \)
7 \( 1 + (-2.82 - 2.82i)T + 7iT^{2} \)
11 \( 1 - 2.94iT - 11T^{2} \)
13 \( 1 - 0.738iT - 13T^{2} \)
17 \( 1 + 6.61T + 17T^{2} \)
19 \( 1 + (-5.03 - 5.03i)T + 19iT^{2} \)
23 \( 1 + 6.39iT - 23T^{2} \)
29 \( 1 + (-2.49 + 2.49i)T - 29iT^{2} \)
31 \( 1 + (1.48 + 1.48i)T + 31iT^{2} \)
41 \( 1 + 7.39iT - 41T^{2} \)
43 \( 1 + 6.98iT - 43T^{2} \)
47 \( 1 + (-7.56 - 7.56i)T + 47iT^{2} \)
53 \( 1 + (-1.94 + 1.94i)T - 53iT^{2} \)
59 \( 1 + (-8.34 - 8.34i)T + 59iT^{2} \)
61 \( 1 + (2.31 + 2.31i)T + 61iT^{2} \)
67 \( 1 + (2.10 - 2.10i)T - 67iT^{2} \)
71 \( 1 + 2.52T + 71T^{2} \)
73 \( 1 + (7.18 + 7.18i)T + 73iT^{2} \)
79 \( 1 + (-5.69 - 5.69i)T + 79iT^{2} \)
83 \( 1 + (2.72 - 2.72i)T - 83iT^{2} \)
89 \( 1 + (-4.25 + 4.25i)T - 89iT^{2} \)
97 \( 1 - 3.40T + 97T^{2} \)
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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

−11.86099999640896797643863327982, −10.59082990858089009694322148147, −9.390596567012117721606510717615, −8.816111507093667245211550495997, −8.273347775874257836855902323212, −7.29956134028833063558553737064, −5.62054558653382708800660973335, −4.61301822557942401863331430062, −4.06222129606110773232734277710, −2.25864601492462132306866495074, 1.17269269816316941208357190051, 2.64337588363334906791953139361, 3.58156354984820673636730456832, 4.83653896653692127989628971677, 6.74985066539254350142102016031, 7.51129147984367589308592090546, 8.192687041408040221061221643882, 9.067930391417668773315638653980, 10.46827069340552966546461783358, 11.29476974062328650866845069864

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