Properties

Label 2-370-37.12-c1-0-0
Degree $2$
Conductor $370$
Sign $-0.230 - 0.973i$
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  + (0.766 − 0.642i)2-s + (−1.90 − 1.59i)3-s + (0.173 − 0.984i)4-s + (−0.939 + 0.342i)5-s − 2.48·6-s + (−4.17 + 1.51i)7-s + (−0.500 − 0.866i)8-s + (0.551 + 3.12i)9-s + (−0.5 + 0.866i)10-s + (2.64 + 4.57i)11-s + (−1.90 + 1.59i)12-s + (−0.291 + 1.65i)13-s + (−2.22 + 3.84i)14-s + (2.33 + 0.849i)15-s + (−0.939 − 0.342i)16-s + (−1.09 − 6.22i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−1.09 − 0.922i)3-s + (0.0868 − 0.492i)4-s + (−0.420 + 0.152i)5-s − 1.01·6-s + (−1.57 + 0.574i)7-s + (−0.176 − 0.306i)8-s + (0.183 + 1.04i)9-s + (−0.158 + 0.273i)10-s + (0.797 + 1.38i)11-s + (−0.549 + 0.461i)12-s + (−0.0809 + 0.458i)13-s + (−0.593 + 1.02i)14-s + (0.602 + 0.219i)15-s + (−0.234 − 0.0855i)16-s + (−0.266 − 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.230 - 0.973i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.230 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0580168 + 0.0733564i\)
\(L(\frac12)\) \(\approx\) \(0.0580168 + 0.0733564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + (5.41 - 2.77i)T \)
good3 \( 1 + (1.90 + 1.59i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (4.17 - 1.51i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-2.64 - 4.57i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.291 - 1.65i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (1.09 + 6.22i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (0.698 + 0.586i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (2.72 - 4.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.39 + 5.88i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.00T + 31T^{2} \)
41 \( 1 + (1.27 - 7.24i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + 2.18T + 43T^{2} \)
47 \( 1 + (1.72 - 2.98i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.95 + 0.710i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (8.25 + 3.00i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.910 + 5.16i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-4.61 + 1.68i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-6.18 - 5.19i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 - 8.95T + 73T^{2} \)
79 \( 1 + (-3.47 + 1.26i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (3.03 + 17.2i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (5.71 + 2.07i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-4.75 + 8.24i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.78247056166479532778716029055, −11.29187647002460385411719814959, −9.751414463425368220491979309407, −9.376945073968584961021239209778, −7.34211049885596498108597431450, −6.76546719843733233714711216385, −6.01647630044507659979845561228, −4.84955061058683504908846016059, −3.48334692065827791487743711091, −1.97178130974407670438710996757, 0.05756900099449319465007910553, 3.64324452276355741562977550812, 3.85989973936150435417898234323, 5.41862303523444359082887577965, 6.15288096031129119116966269507, 6.87547486373305979872538826945, 8.392905575814330320325647002298, 9.360977018231391520093220715775, 10.65262751077131653616002767203, 10.83697976064933602224942737863

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