↑

Properties

Label 2-370-5.4-c1-0-13
Degree $2$
Conductor $370$
Sign $-0.493 + 0.869i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 0.332i·3-s − 4-s + (−1.10 + 1.94i)5-s − 0.332·6-s − 3.51i·7-s + i·8-s + 2.88·9-s + (1.94 + 1.10i)10-s − 0.290·11-s + 0.332i·12-s − 7.12i·13-s − 3.51·14-s + (0.647 + 0.367i)15-s + 16-s − 6.17i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.192i·3-s − 0.5·4-s + (−0.493 + 0.869i)5-s − 0.135·6-s − 1.32i·7-s + 0.353i·8-s + 0.963·9-s + (0.614 + 0.349i)10-s − 0.0874·11-s + 0.0961i·12-s − 1.97i·13-s − 0.938·14-s + (0.167 + 0.0948i)15-s + 0.250·16-s − 1.49i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.493 + 0.869i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.493 + 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.552802 - 0.949482i\)
\(L(\frac12)\) \(\approx\) \(0.552802 - 0.949482i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.10 - 1.94i)T \)
37 \( 1 + iT \)
good3 \( 1 + 0.332iT - 3T^{2} \)
7 \( 1 + 3.51iT - 7T^{2} \)
11 \( 1 + 0.290T + 11T^{2} \)
13 \( 1 + 7.12iT - 13T^{2} \)
17 \( 1 + 6.17iT - 17T^{2} \)
19 \( 1 + 5.83T + 19T^{2} \)
23 \( 1 - 6.45iT - 23T^{2} \)
29 \( 1 - 1.18T + 29T^{2} \)
31 \( 1 - 9.77T + 31T^{2} \)
41 \( 1 - 1.64T + 41T^{2} \)
43 \( 1 - 5.34iT - 43T^{2} \)
47 \( 1 - 5.69iT - 47T^{2} \)
53 \( 1 + 9.32iT - 53T^{2} \)
59 \( 1 + 5.94T + 59T^{2} \)
61 \( 1 + 1.27T + 61T^{2} \)
67 \( 1 - 6.04iT - 67T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 + 1.71iT - 73T^{2} \)
79 \( 1 + 8.25T + 79T^{2} \)
83 \( 1 + 2.41iT - 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 - 15.1iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.91685856606401191423956488224, −10.34174777690985849949499562219, −9.731836019632229587162335765823, −7.990485948980809483054659004001, −7.51888167354996095538774066820, −6.48380175544355719583034637370, −4.84097319987002636865628462562, −3.79464750291226696565121795316, −2.77829908835786729281623290393, −0.789106106064465189110723694796, 1.92885069085282229420940045713, 4.18685234094914153217461541719, 4.62108350795815047941675418271, 6.05865191335474663038292334804, 6.79971351570144976853615607258, 8.303254357412650483255692765780, 8.686448010840243472046606744773, 9.546587263296160700959930066391, 10.71266784880626321082896852138, 12.11874587080776903695938488514

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