↑

Properties

Label 2-370-185.23-c1-0-2
Degree $2$
Conductor $370$
Sign $-0.991 - 0.130i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.658 − 0.176i)3-s + (0.499 + 0.866i)4-s + (−1.32 + 1.80i)5-s + (−0.482 − 0.482i)6-s + (−4.10 − 1.09i)7-s + 0.999i·8-s + (−2.19 − 1.26i)9-s + (−2.04 + 0.901i)10-s + 2.14i·11-s + (−0.176 − 0.658i)12-s + (−4.46 + 2.57i)13-s + (−3.00 − 3.00i)14-s + (1.18 − 0.955i)15-s + (−0.5 + 0.866i)16-s + (1.60 − 2.78i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.380 − 0.101i)3-s + (0.249 + 0.433i)4-s + (−0.590 + 0.806i)5-s + (−0.196 − 0.196i)6-s + (−1.55 − 0.415i)7-s + 0.353i·8-s + (−0.731 − 0.422i)9-s + (−0.647 + 0.285i)10-s + 0.646i·11-s + (−0.0509 − 0.190i)12-s + (−1.23 + 0.714i)13-s + (−0.802 − 0.802i)14-s + (0.307 − 0.246i)15-s + (−0.125 + 0.216i)16-s + (0.389 − 0.675i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.991 - 0.130i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0348199 + 0.533415i\)
\(L(\frac12)\) \(\approx\) \(0.0348199 + 0.533415i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (1.32 - 1.80i)T \)
37 \( 1 + (5.38 + 2.82i)T \)
good3 \( 1 + (0.658 + 0.176i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (4.10 + 1.09i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 - 2.14iT - 11T^{2} \)
13 \( 1 + (4.46 - 2.57i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.60 + 2.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.69 - 1.79i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 2.23iT - 23T^{2} \)
29 \( 1 + (0.485 + 0.485i)T + 29iT^{2} \)
31 \( 1 + (5.21 - 5.21i)T - 31iT^{2} \)
41 \( 1 + (-0.247 + 0.142i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 - 12.4iT - 43T^{2} \)
47 \( 1 + (-2.42 + 2.42i)T - 47iT^{2} \)
53 \( 1 + (7.23 - 1.93i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.59 - 5.94i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.45 - 1.72i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.635 + 2.37i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (6.41 + 11.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.49 + 9.49i)T - 73iT^{2} \)
79 \( 1 + (0.585 + 0.156i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-3.55 + 0.951i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (5.82 - 1.56i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.11913268997779892286442525395, −11.16156735666603114216679021750, −9.944153508693979606386750491295, −9.308575083780005399283178249475, −7.49860268824330631983994812239, −7.08985919962024621273258644263, −6.22533026895216435055689647762, −5.06777368658117657181429314232, −3.61725659196839512308461759193, −2.90792241051043597721056933134, 0.28960832444079380046975925201, 2.77401006575144468690075519326, 3.69745352686632564407848361227, 5.28317797973167673094146574787, 5.63100558738777770467806311962, 7.00870113841013135787335870964, 8.201333069803457188207552829265, 9.305716508973647599370179373281, 10.09369940531019084497670562290, 11.18767706190992762612763645583

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