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Properties

Label 2-370-185.174-c1-0-17
Degree $2$
Conductor $370$
Sign $-0.812 - 0.582i$
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 + (−2.73 − 1.57i)3-s + (0.499 − 0.866i)4-s + (−0.445 − 2.19i)5-s − 3.15·6-s + (−1.45 − 0.837i)7-s − 0.999i·8-s + (3.48 + 6.02i)9-s + (−1.48 − 1.67i)10-s − 4.48·11-s + (−2.73 + 1.57i)12-s + (4.18 + 2.41i)13-s − 1.67·14-s + (−2.24 + 6.69i)15-s + (−0.5 − 0.866i)16-s + (−2.14 + 1.24i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−1.57 − 0.911i)3-s + (0.249 − 0.433i)4-s + (−0.199 − 0.979i)5-s − 1.28·6-s + (−0.548 − 0.316i)7-s − 0.353i·8-s + (1.16 + 2.00i)9-s + (−0.468 − 0.529i)10-s − 1.35·11-s + (−0.789 + 0.455i)12-s + (1.16 + 0.670i)13-s − 0.447·14-s + (−0.578 + 1.72i)15-s + (−0.125 − 0.216i)16-s + (−0.521 + 0.300i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.812 - 0.582i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.812 - 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.166317 + 0.517255i\)
\(L(\frac12)\) \(\approx\) \(0.166317 + 0.517255i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.445 + 2.19i)T \)
37 \( 1 + (-0.920 + 6.01i)T \)
good3 \( 1 + (2.73 + 1.57i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.45 + 0.837i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.48T + 11T^{2} \)
13 \( 1 + (-4.18 - 2.41i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.14 - 1.24i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.28 + 3.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.67iT - 23T^{2} \)
29 \( 1 + 9.08T + 29T^{2} \)
31 \( 1 + 2.83T + 31T^{2} \)
41 \( 1 + (-0.175 + 0.303i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 3.80iT - 43T^{2} \)
47 \( 1 + 2.63iT - 47T^{2} \)
53 \( 1 + (-4.04 + 2.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.91 - 8.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.83 + 11.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.8 + 6.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.80 - 4.86i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.13iT - 73T^{2} \)
79 \( 1 + (-5.04 + 8.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.80 + 3.35i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.27 + 9.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.18iT - 97T^{2} \)
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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

−11.20813644961478734112567552139, −10.41218501836136407008835798971, −9.088611023974161494637625509611, −7.67439270819733867307902424206, −6.81837508912761237099215215674, −5.72185561462553249872546445068, −5.18805102575839679490893907087, −3.96867710850310345550685988870, −1.84057334388803375671819070754, −0.35444599976740946024706756979, 3.08863791339409494222345756402, 4.08096667971452556302674329756, 5.43971846631698443857726579181, 5.88425635877858811169263647767, 6.80688261307716732857493500034, 7.948152746972992718197074319304, 9.562489806430664500877871733034, 10.50905666241614980455162805387, 10.97955132087149450126509120460, 11.75436304440470526686502996543

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