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Properties

Label 2-370-185.104-c1-0-1
Degree $2$
Conductor $370$
Sign $-0.286 + 0.958i$
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 + (−2.07 + 0.834i)5-s − 2.96i·6-s + (1.02 − 0.180i)7-s + (−0.500 + 0.866i)8-s + (−4.42 − 3.71i)9-s + (1.66 − 1.49i)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 + (−0.220 − 6.62i)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 + (−0.927 + 0.373i)5-s − 1.20i·6-s + (0.387 − 0.0682i)7-s + (−0.176 + 0.306i)8-s + (−1.47 − 1.23i)9-s + (0.526 − 0.472i)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 + (−0.0568 − 1.70i)15-s + (0.0434 − 0.246i)16-s + (−0.230 − 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.286 + 0.958i$
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: \((2,\ 370,\ (\ :1/2),\ -0.286 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.110748 - 0.148698i\)
\(L(\frac12)\) \(\approx\) \(0.110748 - 0.148698i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 - 0.834i)T \)
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

−11.42127897472290003988884323273, −11.17355557613811501619421821534, −10.08589606142493605470025489827, −9.547146166360138466697041944328, −8.497587115989259354192908508547, −7.40958691382850605411738870545, −6.41436572721363301129951421389, −4.79829698952146195485825280412, −4.50409597480450318422569146730, −2.85588968109358150154170104824, 0.16708447186483375855978195113, 1.54879309152007002542983890896, 3.09069826930896983896549830653, 5.00893733267461906792699617504, 6.13404045165556524504307899032, 7.27350344651055293525038351964, 7.991884098670990162446017832447, 8.343684211099782084002628692777, 9.956577594269116306254798334081, 11.05034253720854045593010388049

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