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Properties

Label 2-370-185.169-c1-0-13
Degree $2$
Conductor $370$
Sign $0.811 - 0.583i$
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 + (0.489 + 1.34i)3-s + (0.766 + 0.642i)4-s + (0.720 − 2.11i)5-s + 1.42i·6-s + (1.53 + 0.270i)7-s + (0.500 + 0.866i)8-s + (0.731 − 0.614i)9-s + (1.40 − 1.74i)10-s + (−1.42 − 2.46i)11-s + (−0.489 + 1.34i)12-s + (2.83 + 2.38i)13-s + (1.34 + 0.777i)14-s + (3.19 − 0.0664i)15-s + (0.173 + 0.984i)16-s + (−5.50 + 4.61i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.282 + 0.775i)3-s + (0.383 + 0.321i)4-s + (0.322 − 0.946i)5-s + 0.583i·6-s + (0.578 + 0.102i)7-s + (0.176 + 0.306i)8-s + (0.243 − 0.204i)9-s + (0.443 − 0.551i)10-s + (−0.428 − 0.742i)11-s + (−0.141 + 0.387i)12-s + (0.787 + 0.660i)13-s + (0.359 + 0.207i)14-s + (0.825 − 0.0171i)15-s + (0.0434 + 0.246i)16-s + (−1.33 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.811 - 0.583i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.811 - 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22041 + 0.715260i\)
\(L(\frac12)\) \(\approx\) \(2.22041 + 0.715260i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.720 + 2.11i)T \)
37 \( 1 + (5.84 + 1.67i)T \)
good3 \( 1 + (-0.489 - 1.34i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (-1.53 - 0.270i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (1.42 + 2.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.83 - 2.38i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (5.50 - 4.61i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (0.620 + 1.70i)T + (-14.5 + 12.2i)T^{2} \)
23 \( 1 + (1.61 - 2.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.33 - 4.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.60iT - 31T^{2} \)
41 \( 1 + (-3.11 - 2.61i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 + (1.72 + 0.996i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.80 - 1.72i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-4.36 + 0.769i)T + (55.4 - 20.1i)T^{2} \)
61 \( 1 + (-2.54 + 3.02i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-2.50 - 0.441i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-10.6 + 3.88i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + 8.43iT - 73T^{2} \)
79 \( 1 + (1.88 + 0.331i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (5.66 + 6.75i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (5.96 - 1.05i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (2.61 - 4.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.33531718467517721057375656114, −10.82749006406395572038824135862, −9.406655007643451258682694547536, −8.811684725117203078878521430922, −7.914065711077451427727613306210, −6.41653984126629230434022739428, −5.49478311393790655247441584717, −4.42348576102982432820352830128, −3.75767839900461873470000227058, −1.88391949255072737809077716348, 1.83109622058151599273070159739, 2.73702852763771551636282428242, 4.24254241499267744743309946923, 5.44913334325400877217010521573, 6.67331448633629927604627130779, 7.29938183312336605627195483221, 8.255429675614800919428414036701, 9.692895491432691462671984745498, 10.72618036173720177577819194808, 11.18481038817022227022165185346

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