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Properties

Label 2-370-185.104-c1-0-19
Degree $2$
Conductor $370$
Sign $-0.648 + 0.760i$
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  + (0.939 − 0.342i)2-s + (1.01 − 2.78i)3-s + (0.766 − 0.642i)4-s + (−2.23 − 0.0743i)5-s − 2.96i·6-s + (−1.02 + 0.180i)7-s + (0.500 − 0.866i)8-s + (−4.42 − 3.71i)9-s + (−2.12 + 0.694i)10-s + (−1.08 + 1.88i)11-s + (−1.01 − 2.78i)12-s + (3.72 − 3.12i)13-s + (−0.900 + 0.519i)14-s + (−2.47 + 6.14i)15-s + (0.173 − 0.984i)16-s + (0.949 + 0.796i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.585 − 1.60i)3-s + (0.383 − 0.321i)4-s + (−0.999 − 0.0332i)5-s − 1.20i·6-s + (−0.387 + 0.0682i)7-s + (0.176 − 0.306i)8-s + (−1.47 − 1.23i)9-s + (−0.672 + 0.219i)10-s + (−0.328 + 0.569i)11-s + (−0.292 − 0.803i)12-s + (1.03 − 0.867i)13-s + (−0.240 + 0.138i)14-s + (−0.638 + 1.58i)15-s + (0.0434 − 0.246i)16-s + (0.230 + 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.648 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.648 + 0.760i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.648 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.794649 - 1.72187i\)
\(L(\frac12)\) \(\approx\) \(0.794649 - 1.72187i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (2.23 + 0.0743i)T \)
37 \( 1 + (5.16 - 3.21i)T \)
good3 \( 1 + (-1.01 + 2.78i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (1.02 - 0.180i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (1.08 - 1.88i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.72 + 3.12i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.949 - 0.796i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-1.59 + 4.38i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (-2.09 - 3.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.14 - 3.54i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.83iT - 31T^{2} \)
41 \( 1 + (5.14 - 4.32i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 2.74T + 43T^{2} \)
47 \( 1 + (-10.9 + 6.29i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.66 - 1.70i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (5.46 + 0.962i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (-1.79 - 2.13i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-8.75 + 1.54i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-6.94 - 2.52i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 7.86iT - 73T^{2} \)
79 \( 1 + (15.6 - 2.76i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (2.02 - 2.40i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (12.0 + 2.13i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-5.15 - 8.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.51648703477202797034412882845, −10.40644766769460091930190577222, −8.880360396849568069637909308721, −8.050278609679293329688752379093, −7.22068238446022007787420208324, −6.51180418084238276833370142561, −5.20076053536801597108281868637, −3.56941703334736490014042706748, −2.72909138123680444528070534345, −1.07190434587939176760437348206, 3.02654084526926575538005730625, 3.75786697646422989940286707624, 4.49598907600492055922726855165, 5.63767913073229379494877137727, 6.94625442330641246656629465898, 8.333378933201604360673452769409, 8.738365516288800992197132557126, 10.04581242394201173848326338837, 10.80509501551450232147458832055, 11.58924469262655441161516771205

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