Properties

Label 2-370-185.84-c1-0-3
Degree $2$
Conductor $370$
Sign $0.232 - 0.972i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−2 − i)5-s + (−0.633 + 0.366i)7-s − 0.999i·8-s + (−1.5 + 2.59i)9-s + (1.23 + 1.86i)10-s + 4.73·11-s + (−4.73 + 2.73i)13-s + 0.732·14-s + (−0.5 + 0.866i)16-s + (1.5 + 0.866i)17-s + (2.59 − 1.5i)18-s + (2.36 + 4.09i)19-s + (−0.133 − 2.23i)20-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.894 − 0.447i)5-s + (−0.239 + 0.138i)7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + (0.389 + 0.590i)10-s + 1.42·11-s + (−1.31 + 0.757i)13-s + 0.195·14-s + (−0.125 + 0.216i)16-s + (0.363 + 0.210i)17-s + (0.612 − 0.353i)18-s + (0.542 + 0.940i)19-s + (−0.0299 − 0.499i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.232 - 0.972i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.232 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.463987 + 0.366292i\)
\(L(\frac12)\) \(\approx\) \(0.463987 + 0.366292i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
37 \( 1 + (-2.59 + 5.5i)T \)
good3 \( 1 + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.633 - 0.366i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
13 \( 1 + (4.73 - 2.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 0.866i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.36 - 4.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.46iT - 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
31 \( 1 + 9.66T + 31T^{2} \)
41 \( 1 + (-3.96 - 6.86i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 5.26iT - 43T^{2} \)
47 \( 1 + 3.26iT - 47T^{2} \)
53 \( 1 + (-10.7 - 6.19i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.26 - 2.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.13 + 3.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.56 - 2.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.56 + 13.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 15.8iT - 73T^{2} \)
79 \( 1 + (0.830 + 1.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.19 + 4.73i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.232 + 0.401i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.19iT - 97T^{2} \)
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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

−11.73424132914641201495268432537, −10.79136132314020974657888789873, −9.467616822821188254671845324163, −9.089968059633722108883901365732, −7.77447218230762367604237294188, −7.32977481453183313854146117563, −5.79976790864534711134385056235, −4.45216539537335777350910946237, −3.36686209139285066436103425853, −1.68470661291711487312297580928, 0.49556059449336150711849857267, 2.85757778795856696267194885522, 4.03824625796922037485299798927, 5.52621835129458002959736948540, 6.85399168041481655328300829744, 7.21434555852360978173304763714, 8.485157530115377498949894357400, 9.280226734984314897496876297338, 10.13984436302712989873762318250, 11.26794618710815542064642728996

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