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Properties

Label 2-370-37.34-c1-0-13
Degree $2$
Conductor $370$
Sign $0.915 + 0.402i$
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 + (2.61 − 2.19i)3-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s + 3.40·6-s + (−2.86 − 1.04i)7-s + (−0.500 + 0.866i)8-s + (1.49 − 8.48i)9-s + (0.5 + 0.866i)10-s + (−0.334 + 0.578i)11-s + (2.61 + 2.19i)12-s + (0.343 + 1.94i)13-s + (−1.52 − 2.64i)14-s + (3.20 − 1.16i)15-s + (−0.939 + 0.342i)16-s + (−0.441 + 2.50i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (1.50 − 1.26i)3-s + (0.0868 + 0.492i)4-s + (0.420 + 0.152i)5-s + 1.39·6-s + (−1.08 − 0.394i)7-s + (−0.176 + 0.306i)8-s + (0.498 − 2.82i)9-s + (0.158 + 0.273i)10-s + (−0.100 + 0.174i)11-s + (0.753 + 0.632i)12-s + (0.0953 + 0.540i)13-s + (−0.408 − 0.706i)14-s + (0.827 − 0.301i)15-s + (−0.234 + 0.0855i)16-s + (−0.107 + 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.915 + 0.402i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.915 + 0.402i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60569 - 0.547583i\)
\(L(\frac12)\) \(\approx\) \(2.60569 - 0.547583i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.70 + 4.82i)T \)
good3 \( 1 + (-2.61 + 2.19i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (2.86 + 1.04i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (0.334 - 0.578i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.343 - 1.94i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.441 - 2.50i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (2.35 - 1.97i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (-3.51 - 6.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.08 - 5.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.00T + 31T^{2} \)
41 \( 1 + (0.522 + 2.96i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + 9.29T + 43T^{2} \)
47 \( 1 + (2.65 + 4.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-13.3 + 4.85i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (4.36 - 1.59i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.96 - 11.1i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (4.94 + 1.79i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (2.12 - 1.78i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 8.43T + 73T^{2} \)
79 \( 1 + (12.2 + 4.46i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-3.14 + 17.8i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (3.15 - 1.14i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (0.608 + 1.05i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.78852972746858308357182283937, −10.13372006314957062668809773646, −9.190865087897040432486736727478, −8.454506645130339559217068518206, −7.28148516259864780434633697802, −6.84630715531414582961455227737, −5.91771542019366752009385450141, −3.88511719089573828028851799007, −3.09012165953213639872546846145, −1.79795268058372365828294890490, 2.52636118369327934221024984294, 3.05508215931545923673334541632, 4.26727795758958753616954672302, 5.18684727438275976175433152835, 6.53517267608777555635786515176, 8.114915189984649436484086144540, 8.979165218431491117023197985353, 9.706976057044936797295142343793, 10.25533664312925939350989774419, 11.22007221057454109709337449879

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