↑

Properties

Label 2-370-37.33-c1-0-13
Degree $2$
Conductor $370$
Sign $-0.467 - 0.884i$
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  + (0.173 − 0.984i)2-s + (−0.471 − 2.67i)3-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)5-s − 2.71·6-s + (−2.35 + 1.97i)7-s + (−0.5 + 0.866i)8-s + (−4.09 + 1.49i)9-s + (0.5 + 0.866i)10-s + (0.806 − 1.39i)11-s + (−0.471 + 2.67i)12-s + (−2.63 − 0.960i)13-s + (1.53 + 2.66i)14-s + (2.07 + 1.74i)15-s + (0.766 + 0.642i)16-s + (−6.59 + 2.39i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.272 − 1.54i)3-s + (−0.469 − 0.171i)4-s + (−0.342 + 0.287i)5-s − 1.10·6-s + (−0.890 + 0.746i)7-s + (−0.176 + 0.306i)8-s + (−1.36 + 0.497i)9-s + (0.158 + 0.273i)10-s + (0.243 − 0.421i)11-s + (−0.136 + 0.771i)12-s + (−0.731 − 0.266i)13-s + (0.410 + 0.711i)14-s + (0.536 + 0.450i)15-s + (0.191 + 0.160i)16-s + (−1.59 + 0.581i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.467 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.467 - 0.884i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.467 - 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.213349 + 0.354099i\)
\(L(\frac12)\) \(\approx\) \(0.213349 + 0.354099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-3.90 - 4.66i)T \)
good3 \( 1 + (0.471 + 2.67i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (2.35 - 1.97i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.806 + 1.39i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.63 + 0.960i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (6.59 - 2.39i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (0.994 + 5.63i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (-1.56 - 2.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.19 + 7.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.98T + 31T^{2} \)
41 \( 1 + (8.76 + 3.19i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + 5.86T + 43T^{2} \)
47 \( 1 + (1.49 + 2.58i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.0 + 8.45i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (1.51 + 1.26i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (7.12 + 2.59i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-5.33 + 4.47i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.264 - 1.50i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + 4.15T + 73T^{2} \)
79 \( 1 + (-1.91 + 1.60i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-8.44 + 3.07i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (1.48 + 1.24i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-6.09 - 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.23951651047381406670731061769, −9.950084657967512473682475532950, −8.857470777820198934893957138297, −8.001845487072016556936710748681, −6.64226948869133157049902431154, −6.33747786364233811592250945268, −4.80357073225514365771298594254, −3.06013996182583699022134718508, −2.15523687666187439782001385947, −0.26341190818668655379049192550, 3.28076910473945334116649048154, 4.42124234908025618087063822483, 4.78887112147127627775510994454, 6.29109345379800766550919773434, 7.11634862981842937397007520426, 8.490611518138453417809334910775, 9.383808495082948730988952561109, 10.03684358767010824860062063024, 10.82443376015221728434759757226, 11.98646619796477734195283796287

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