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Properties

Label 2-370-185.169-c1-0-11
Degree $2$
Conductor $370$
Sign $0.999 + 0.00810i$
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  + (0.939 + 0.342i)2-s + (−0.362 − 0.995i)3-s + (0.766 + 0.642i)4-s + (2.12 + 0.683i)5-s − 1.05i·6-s + (−1.06 − 0.187i)7-s + (0.500 + 0.866i)8-s + (1.43 − 1.20i)9-s + (1.76 + 1.37i)10-s + (0.436 + 0.755i)11-s + (0.362 − 0.995i)12-s + (1.18 + 0.992i)13-s + (−0.933 − 0.539i)14-s + (−0.0906 − 2.36i)15-s + (0.173 + 0.984i)16-s + (−0.626 + 0.525i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (−0.209 − 0.574i)3-s + (0.383 + 0.321i)4-s + (0.952 + 0.305i)5-s − 0.432i·6-s + (−0.401 − 0.0707i)7-s + (0.176 + 0.306i)8-s + (0.479 − 0.402i)9-s + (0.558 + 0.433i)10-s + (0.131 + 0.227i)11-s + (0.104 − 0.287i)12-s + (0.328 + 0.275i)13-s + (−0.249 − 0.144i)14-s + (−0.0234 − 0.611i)15-s + (0.0434 + 0.246i)16-s + (−0.151 + 0.127i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.999 + 0.00810i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.999 + 0.00810i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12468 - 0.00861188i\)
\(L(\frac12)\) \(\approx\) \(2.12468 - 0.00861188i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (-2.12 - 0.683i)T \)
37 \( 1 + (4.86 - 3.64i)T \)
good3 \( 1 + (0.362 + 0.995i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (1.06 + 0.187i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-0.436 - 0.755i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.18 - 0.992i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.626 - 0.525i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (0.570 + 1.56i)T + (-14.5 + 12.2i)T^{2} \)
23 \( 1 + (-2.32 + 4.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.84 - 1.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.18iT - 31T^{2} \)
41 \( 1 + (-1.27 - 1.06i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + 2.12T + 43T^{2} \)
47 \( 1 + (4.23 + 2.44i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.91 + 0.865i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (10.8 - 1.90i)T + (55.4 - 20.1i)T^{2} \)
61 \( 1 + (-1.22 + 1.45i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (5.75 + 1.01i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (10.2 - 3.71i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 + (4.86 + 0.858i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-2.61 - 3.11i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (15.0 - 2.66i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (-5.51 + 9.55i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.57544215346053582079091316512, −10.54124513519936248930016654757, −9.639193398069745548093031855046, −8.591309027894922565721656196857, −7.06831808905052333947076254965, −6.66484214711137428311084900790, −5.77499185710800774728414735136, −4.52097489333390051872455789509, −3.12288765424161898488161071604, −1.68727980633867582026239196669, 1.73721812485559845733912831369, 3.27212791262015223328164673938, 4.50987319611229669507191326991, 5.46412948375066257486489897270, 6.22933570685471172164154114155, 7.47861830419772272369952693156, 8.949282466589652391330715190052, 9.768881452152524971025998677044, 10.49541113450325090884686228945, 11.29976959097160963001327500348

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