Properties

Label 2-370-185.183-c1-0-12
Degree $2$
Conductor $370$
Sign $0.999 + 0.0377i$
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  + (0.642 + 0.766i)2-s + (−1.69 + 0.147i)3-s + (−0.173 + 0.984i)4-s + (1.83 − 1.27i)5-s + (−1.19 − 1.19i)6-s + (1.37 − 2.94i)7-s + (−0.866 + 0.500i)8-s + (−0.118 + 0.0208i)9-s + (2.15 + 0.582i)10-s + (3.04 − 1.76i)11-s + (0.147 − 1.69i)12-s + (−1.37 − 0.243i)13-s + (3.13 − 0.840i)14-s + (−2.91 + 2.43i)15-s + (−0.939 − 0.342i)16-s + (−0.343 − 1.94i)17-s + ⋯
L(s)  = 1  + (0.454 + 0.541i)2-s + (−0.976 + 0.0853i)3-s + (−0.0868 + 0.492i)4-s + (0.820 − 0.571i)5-s + (−0.489 − 0.489i)6-s + (0.518 − 1.11i)7-s + (−0.306 + 0.176i)8-s + (−0.0394 + 0.00696i)9-s + (0.682 + 0.184i)10-s + (0.919 − 0.530i)11-s + (0.0426 − 0.488i)12-s + (−0.382 − 0.0674i)13-s + (0.838 − 0.224i)14-s + (−0.751 + 0.628i)15-s + (−0.234 − 0.0855i)16-s + (−0.0833 − 0.472i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.999 + 0.0377i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.999 + 0.0377i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46551 - 0.0276399i\)
\(L(\frac12)\) \(\approx\) \(1.46551 - 0.0276399i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 - 0.766i)T \)
5 \( 1 + (-1.83 + 1.27i)T \)
37 \( 1 + (2.06 - 5.72i)T \)
good3 \( 1 + (1.69 - 0.147i)T + (2.95 - 0.520i)T^{2} \)
7 \( 1 + (-1.37 + 2.94i)T + (-4.49 - 5.36i)T^{2} \)
11 \( 1 + (-3.04 + 1.76i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.37 + 0.243i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.343 + 1.94i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (-7.25 + 0.634i)T + (18.7 - 3.29i)T^{2} \)
23 \( 1 + (-2.79 - 1.61i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.05 - 7.65i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (2.70 - 2.70i)T - 31iT^{2} \)
41 \( 1 + (1.78 + 0.315i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + 3.81iT - 43T^{2} \)
47 \( 1 + (2.51 + 9.39i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.16 + 6.79i)T + (-34.0 + 40.6i)T^{2} \)
59 \( 1 + (0.319 - 0.148i)T + (37.9 - 45.1i)T^{2} \)
61 \( 1 + (3.68 + 5.26i)T + (-20.8 + 57.3i)T^{2} \)
67 \( 1 + (14.3 + 6.69i)T + (43.0 + 51.3i)T^{2} \)
71 \( 1 + (-12.2 - 10.2i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (3.39 - 3.39i)T - 73iT^{2} \)
79 \( 1 + (1.02 - 2.19i)T + (-50.7 - 60.5i)T^{2} \)
83 \( 1 + (4.24 - 6.06i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (-6.07 - 13.0i)T + (-57.2 + 68.1i)T^{2} \)
97 \( 1 + (1.59 - 2.76i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.50299073997229824509548214337, −10.64424757884641907195831953951, −9.550545102605215050246048437878, −8.600030457730472759215460212949, −7.28322651666772208972999834723, −6.53580233718057559308771391906, −5.25830511970516025678126417210, −5.01027342840490778267301064762, −3.45783084221940335720533065856, −1.13255759421956534047368256773, 1.68144613731945845500996405024, 2.94312432957277477708356592041, 4.65339275617410850710832217687, 5.69375620889480557597299071100, 6.12096959889936080982037788181, 7.37688416791745073593837036783, 9.040017515985167563853779515392, 9.677943961742251092877338012654, 10.77695201414556510472347093827, 11.58267856559500316839844598426

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