↑

Properties

Label 2-370-185.103-c1-0-5
Degree $2$
Conductor $370$
Sign $0.569 - 0.821i$
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  + (0.5 + 0.866i)2-s + (−0.676 − 2.52i)3-s + (−0.499 + 0.866i)4-s + (1.00 + 1.99i)5-s + (1.84 − 1.84i)6-s + (1.06 + 3.97i)7-s − 0.999·8-s + (−3.32 + 1.92i)9-s + (−1.22 + 1.86i)10-s + 1.28i·11-s + (2.52 + 0.676i)12-s + (−0.158 + 0.274i)13-s + (−2.90 + 2.90i)14-s + (4.36 − 3.89i)15-s + (−0.5 − 0.866i)16-s + (0.887 − 0.512i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.390 − 1.45i)3-s + (−0.249 + 0.433i)4-s + (0.449 + 0.893i)5-s + (0.754 − 0.754i)6-s + (0.402 + 1.50i)7-s − 0.353·8-s + (−1.10 + 0.640i)9-s + (−0.388 + 0.591i)10-s + 0.386i·11-s + (0.729 + 0.195i)12-s + (−0.0438 + 0.0759i)13-s + (−0.777 + 0.777i)14-s + (1.12 − 1.00i)15-s + (−0.125 − 0.216i)16-s + (0.215 − 0.124i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.569 - 0.821i$
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: \((2,\ 370,\ (\ :1/2),\ 0.569 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32677 + 0.694396i\)
\(L(\frac12)\) \(\approx\) \(1.32677 + 0.694396i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.00 - 1.99i)T \)
37 \( 1 + (-3.18 + 5.18i)T \)
good3 \( 1 + (0.676 + 2.52i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.06 - 3.97i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 1.28iT - 11T^{2} \)
13 \( 1 + (0.158 - 0.274i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.887 + 0.512i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.85 + 0.764i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 9.01T + 23T^{2} \)
29 \( 1 + (5.62 - 5.62i)T - 29iT^{2} \)
31 \( 1 + (-0.661 - 0.661i)T + 31iT^{2} \)
41 \( 1 + (9.70 + 5.60i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 5.24T + 43T^{2} \)
47 \( 1 + (-2.65 + 2.65i)T - 47iT^{2} \)
53 \( 1 + (-1.83 + 6.83i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.26 + 8.44i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (7.25 - 1.94i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.39 - 0.374i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.61 - 6.26i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.58 - 7.58i)T - 73iT^{2} \)
79 \( 1 + (-4.64 + 1.24i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (0.409 - 1.52i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-1.06 - 0.284i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 11.2iT - 97T^{2} \)
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

−11.74706531246970156194192684646, −11.02165592496107703647260757329, −9.440889279445398530752891637694, −8.544123106916516641935354463827, −7.28502592882578273132884466862, −6.94646937495504629941795197224, −5.77778979679607003031498640595, −5.26636106272554202143969111067, −3.01232373581278185120032151871, −1.89997710837387446340857782046, 1.06552374074547639250777494647, 3.34571061098328605187704012563, 4.38709278355045477239462343068, 4.92817392550888857153758140344, 5.94885682880696596061510577873, 7.60139242713731425831980061319, 8.939621462985447043146216047506, 9.703246710002772656217255489074, 10.41372683238554840333183895985, 11.06781279783629420879274939658

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