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Properties

Label 2-370-185.104-c1-0-13
Degree $2$
Conductor $370$
Sign $0.403 + 0.914i$
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 + (0.400 − 1.10i)3-s + (0.766 − 0.642i)4-s + (−0.822 − 2.07i)5-s + 1.17i·6-s + (3.81 − 0.672i)7-s + (−0.500 + 0.866i)8-s + (1.24 + 1.04i)9-s + (1.48 + 1.67i)10-s + (−1.57 + 2.73i)11-s + (−0.400 − 1.10i)12-s + (1.40 − 1.17i)13-s + (−3.35 + 1.93i)14-s + (−2.61 + 0.0716i)15-s + (0.173 − 0.984i)16-s + (−4.46 − 3.74i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.231 − 0.635i)3-s + (0.383 − 0.321i)4-s + (−0.367 − 0.929i)5-s + 0.477i·6-s + (1.44 − 0.254i)7-s + (−0.176 + 0.306i)8-s + (0.416 + 0.349i)9-s + (0.469 + 0.529i)10-s + (−0.476 + 0.825i)11-s + (−0.115 − 0.317i)12-s + (0.389 − 0.326i)13-s + (−0.896 + 0.517i)14-s + (−0.675 + 0.0185i)15-s + (0.0434 − 0.246i)16-s + (−1.08 − 0.908i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.403 + 0.914i$
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: \((2,\ 370,\ (\ :1/2),\ 0.403 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.966980 - 0.630429i\)
\(L(\frac12)\) \(\approx\) \(0.966980 - 0.630429i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.822 + 2.07i)T \)
37 \( 1 + (5.80 + 1.80i)T \)
good3 \( 1 + (-0.400 + 1.10i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (-3.81 + 0.672i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (1.57 - 2.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.40 + 1.17i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (4.46 + 3.74i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-2.67 + 7.36i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (0.0262 + 0.0453i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.42 - 1.97i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.88iT - 31T^{2} \)
41 \( 1 + (-2.75 + 2.30i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 2.80T + 43T^{2} \)
47 \( 1 + (9.42 - 5.44i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-12.8 - 2.25i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-12.1 - 2.13i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (-0.0728 - 0.0867i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (4.55 - 0.803i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (12.4 + 4.52i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 4.68iT - 73T^{2} \)
79 \( 1 + (5.72 - 1.01i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.100 - 0.119i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-2.80 - 0.493i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-8.58 - 14.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.25405299159133976774297558727, −10.32142202456223859438620550750, −9.029037217990942661628778415248, −8.412548255977344607215051981176, −7.48795124167143454396000540605, −7.03022162098182473912832533017, −5.09184022539829798399677701454, −4.63459998677372287533252018930, −2.28613505119670152909331923335, −1.06589462739155474438159066721, 1.81013326096487212169949655896, 3.36283679114922850664636325709, 4.31076573604547779823843770199, 5.85444566358415927966452744958, 7.03611333186100911068321055422, 8.240298203689834128493820960975, 8.536963662261877030814553770238, 10.00073776361092620657816907421, 10.54679465509251843419695932294, 11.38683522007140539924936791557

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