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Properties

Label 2-370-185.103-c1-0-16
Degree $2$
Conductor $370$
Sign $-0.993 - 0.116i$
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  + (−0.5 − 0.866i)2-s + (−0.758 − 2.83i)3-s + (−0.499 + 0.866i)4-s + (2.23 + 0.0488i)5-s + (−2.07 + 2.07i)6-s + (−0.644 − 2.40i)7-s + 0.999·8-s + (−4.84 + 2.79i)9-s + (−1.07 − 1.96i)10-s − 2.91i·11-s + (2.83 + 0.758i)12-s + (1.47 − 2.54i)13-s + (−1.76 + 1.76i)14-s + (−1.55 − 6.36i)15-s + (−0.5 − 0.866i)16-s + (−3.89 + 2.25i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.438 − 1.63i)3-s + (−0.249 + 0.433i)4-s + (0.999 + 0.0218i)5-s + (−0.846 + 0.846i)6-s + (−0.243 − 0.909i)7-s + 0.353·8-s + (−1.61 + 0.932i)9-s + (−0.340 − 0.619i)10-s − 0.878i·11-s + (0.817 + 0.219i)12-s + (0.407 − 0.706i)13-s + (−0.470 + 0.470i)14-s + (−0.402 − 1.64i)15-s + (−0.125 − 0.216i)16-s + (−0.945 + 0.545i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.993 - 0.116i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0555501 + 0.953699i\)
\(L(\frac12)\) \(\approx\) \(0.0555501 + 0.953699i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-2.23 - 0.0488i)T \)
37 \( 1 + (-2.51 - 5.53i)T \)
good3 \( 1 + (0.758 + 2.83i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.644 + 2.40i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 2.91iT - 11T^{2} \)
13 \( 1 + (-1.47 + 2.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.89 - 2.25i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.54 - 0.949i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 6.99T + 23T^{2} \)
29 \( 1 + (3.87 - 3.87i)T - 29iT^{2} \)
31 \( 1 + (-3.48 - 3.48i)T + 31iT^{2} \)
41 \( 1 + (4.86 + 2.81i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 9.87T + 43T^{2} \)
47 \( 1 + (-6.90 + 6.90i)T - 47iT^{2} \)
53 \( 1 + (-2.56 + 9.56i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.43 - 5.34i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.95 + 1.32i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (3.36 - 0.901i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.12 + 14.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.91 + 1.91i)T - 73iT^{2} \)
79 \( 1 + (-4.93 + 1.32i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-2.48 + 9.27i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-5.84 - 1.56i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 1.22iT - 97T^{2} \)
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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

−10.79971245060857352001199610297, −10.44427969000285881431365618957, −8.894216822305488481442791998991, −8.211682409408098328662483666586, −6.92628323852528278625050600362, −6.42874989698329017826208311319, −5.23738451505446410672931455715, −3.28776261361521060631460829226, −1.91328394357270392097406505720, −0.77784879515372244757471272205, 2.44454503270605253382489617913, 4.29478923519073251775025133481, 5.08518341372492437844206768259, 5.98685652350658506262433296667, 6.83495785247823697137743660784, 8.696858211738579043784644269704, 9.294624274566049391374884222535, 9.725701538246112377574087351744, 10.75214304167900995239577198372, 11.46621150796145557723147219556

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