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Properties

Label 2-370-185.103-c1-0-3
Degree $2$
Conductor $370$
Sign $-0.631 - 0.775i$
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 + (0.0936 + 0.349i)3-s + (−0.499 + 0.866i)4-s + (−0.693 + 2.12i)5-s + (−0.255 + 0.255i)6-s + (0.0100 + 0.0374i)7-s − 0.999·8-s + (2.48 − 1.43i)9-s + (−2.18 + 0.462i)10-s + 3.45i·11-s + (−0.349 − 0.0936i)12-s + (−1.24 + 2.15i)13-s + (−0.0273 + 0.0273i)14-s + (−0.807 − 0.0431i)15-s + (−0.5 − 0.866i)16-s + (−1.63 + 0.941i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.0540 + 0.201i)3-s + (−0.249 + 0.433i)4-s + (−0.309 + 0.950i)5-s + (−0.104 + 0.104i)6-s + (0.00378 + 0.0141i)7-s − 0.353·8-s + (0.828 − 0.478i)9-s + (−0.691 + 0.146i)10-s + 1.04i·11-s + (−0.100 − 0.0270i)12-s + (−0.345 + 0.597i)13-s + (−0.00732 + 0.00732i)14-s + (−0.208 − 0.0111i)15-s + (−0.125 − 0.216i)16-s + (−0.395 + 0.228i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.631 - 0.775i$
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: \((2,\ 370,\ (\ :1/2),\ -0.631 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.617896 + 1.30049i\)
\(L(\frac12)\) \(\approx\) \(0.617896 + 1.30049i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.693 - 2.12i)T \)
37 \( 1 + (6.07 + 0.307i)T \)
good3 \( 1 + (-0.0936 - 0.349i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.0100 - 0.0374i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 3.45iT - 11T^{2} \)
13 \( 1 + (1.24 - 2.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.63 - 0.941i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.26 - 1.40i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 0.905T + 23T^{2} \)
29 \( 1 + (-6.45 + 6.45i)T - 29iT^{2} \)
31 \( 1 + (-4.37 - 4.37i)T + 31iT^{2} \)
41 \( 1 + (-8.36 - 4.82i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.19T + 43T^{2} \)
47 \( 1 + (-5.31 + 5.31i)T - 47iT^{2} \)
53 \( 1 + (-0.879 + 3.28i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.03 - 3.85i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.0232 - 0.00623i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (14.8 - 3.98i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-6.17 + 10.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.60 + 6.60i)T - 73iT^{2} \)
79 \( 1 + (-4.82 + 1.29i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (3.63 - 13.5i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-2.48 - 0.666i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 2.55iT - 97T^{2} \)
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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

−11.99350614029554072978939681357, −10.64975373842855049737603759658, −9.978286328451172627859580994121, −8.907953046842780522812979486071, −7.70950794944140737496120468930, −6.87647956350922686366356493113, −6.28241483082517750677995554885, −4.58033858070774219225861116371, −3.97354524997765234380404425467, −2.39096227883825463898128043140, 0.917407241619023335184665623205, 2.55652643766222506208043070395, 4.08733496727011626064329744802, 4.89549099093571661538372725790, 6.02778343369315268796377728099, 7.39026119591096240831350655103, 8.450358471738440968226427722287, 9.164964937569596128800080702503, 10.43532032339206365990003896615, 11.01122445371893837119815677139

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