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Properties

Label 2-370-185.159-c1-0-0
Degree $2$
Conductor $370$
Sign $0.176 - 0.984i$
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 + (1.86 + 1.22i)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 + (−2.07 − 4.13i)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 + (0.835 + 0.549i)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 + (−0.535 − 1.06i)15-s + (−0.125 − 0.216i)16-s + (0.532 + 0.921i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.176 - 0.984i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.176 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.251343 + 0.210177i\)
\(L(\frac12)\) \(\approx\) \(0.251343 + 0.210177i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 - 1.22i)T \)
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

−11.48979776634640674143043998099, −10.65350696027908898319744431882, −10.08652577083789409075248843448, −9.116611184151163808327680581291, −7.68230743334778087301268330814, −6.79554601144037968138177445659, −5.94538395599583214254548965752, −4.82963570929187985719619733011, −3.05309079257975503027984822955, −1.75254525440751499284745569462, 0.26773685226389520832835953721, 2.70839812189742655040705999273, 4.92477152056105443132436780004, 5.33555967970434894447403036522, 6.03722184234071517749448754897, 7.42755989354573768557176067389, 8.388623724694476190386588626003, 9.670903817362866515883807032597, 10.11752520472225823633087540706, 10.82974102087862765915389249033

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