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Properties

Label 2-370-185.64-c1-0-17
Degree $2$
Conductor $370$
Sign $0.0608 + 0.998i$
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  + (0.5 − 0.866i)2-s + (1.79 − 1.03i)3-s + (−0.499 − 0.866i)4-s + (1.99 − 1.00i)5-s − 2.06i·6-s + (1.25 − 0.725i)7-s − 0.999·8-s + (0.638 − 1.10i)9-s + (0.130 − 2.23i)10-s − 5.11·11-s + (−1.79 − 1.03i)12-s + (2.79 + 4.84i)13-s − 1.45i·14-s + (2.54 − 3.86i)15-s + (−0.5 + 0.866i)16-s + (−2.19 + 3.80i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (1.03 − 0.596i)3-s + (−0.249 − 0.433i)4-s + (0.893 − 0.448i)5-s − 0.844i·6-s + (0.474 − 0.274i)7-s − 0.353·8-s + (0.212 − 0.368i)9-s + (0.0413 − 0.705i)10-s − 1.54·11-s + (−0.516 − 0.298i)12-s + (0.775 + 1.34i)13-s − 0.387i·14-s + (0.656 − 0.997i)15-s + (−0.125 + 0.216i)16-s + (−0.532 + 0.921i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0608 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0608 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.0608 + 0.998i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.0608 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71278 - 1.61157i\)
\(L(\frac12)\) \(\approx\) \(1.71278 - 1.61157i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.99 + 1.00i)T \)
37 \( 1 + (-0.667 + 6.04i)T \)
good3 \( 1 + (-1.79 + 1.03i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.25 + 0.725i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 5.11T + 11T^{2} \)
13 \( 1 + (-2.79 - 4.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.19 - 3.80i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.736 - 0.425i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.02T + 23T^{2} \)
29 \( 1 - 3.13iT - 29T^{2} \)
31 \( 1 + 6.98iT - 31T^{2} \)
41 \( 1 + (5.10 + 8.84i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 3.37T + 43T^{2} \)
47 \( 1 - 3.87iT - 47T^{2} \)
53 \( 1 + (-9.19 - 5.30i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.97 - 1.14i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.743 - 0.429i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.48 + 5.47i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.71 + 8.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 + (5.04 - 2.91i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.0647 + 0.0373i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-12.1 - 6.99i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.26T + 97T^{2} \)
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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

−11.04511310295028154055783448654, −10.40803378034916027856048621987, −9.181824152245541333931057140958, −8.566537974940951837622286941019, −7.61799774677143613672069094420, −6.27837572322901815651390078617, −5.15684423151654868919069025401, −3.95353597022706970669004707334, −2.39616603133974675002026766092, −1.72644212642618854903264326448, 2.53976605369694007338187521746, 3.27502055199344352323306096304, 4.90002759560546124507440838448, 5.63335548702366680044480275006, 6.87744868719631774921045313026, 8.193621011692395215483279066023, 8.517928955799831368128188486144, 9.823563956284727607524546293077, 10.37512521450204654439994929029, 11.53540388076746514614202506472

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