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Properties

Label 2-370-185.84-c1-0-14
Degree $2$
Conductor $370$
Sign $0.986 - 0.162i$
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 + (1.41 − 0.815i)3-s + (0.499 + 0.866i)4-s + (2.14 − 0.618i)5-s + 1.63·6-s + (−0.466 + 0.269i)7-s + 0.999i·8-s + (−0.170 + 0.294i)9-s + (2.17 + 0.539i)10-s − 0.829·11-s + (1.41 + 0.815i)12-s + (−0.945 + 0.545i)13-s − 0.539·14-s + (2.53 − 2.62i)15-s + (−0.5 + 0.866i)16-s + (1.01 + 0.585i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.815 − 0.470i)3-s + (0.249 + 0.433i)4-s + (0.961 − 0.276i)5-s + 0.665·6-s + (−0.176 + 0.101i)7-s + 0.353i·8-s + (−0.0566 + 0.0981i)9-s + (0.686 + 0.170i)10-s − 0.250·11-s + (0.407 + 0.235i)12-s + (−0.262 + 0.151i)13-s − 0.144·14-s + (0.653 − 0.677i)15-s + (−0.125 + 0.216i)16-s + (0.245 + 0.141i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.986 - 0.162i$
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: \((2,\ 370,\ (\ :1/2),\ 0.986 - 0.162i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.55535 + 0.209489i\)
\(L(\frac12)\) \(\approx\) \(2.55535 + 0.209489i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.14 + 0.618i)T \)
37 \( 1 + (-4.29 - 4.30i)T \)
good3 \( 1 + (-1.41 + 0.815i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.466 - 0.269i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.829T + 11T^{2} \)
13 \( 1 + (0.945 - 0.545i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.01 - 0.585i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.87 + 6.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.539iT - 23T^{2} \)
29 \( 1 + 9.57T + 29T^{2} \)
31 \( 1 - 3.09T + 31T^{2} \)
41 \( 1 + (0.960 + 1.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 1.29iT - 43T^{2} \)
47 \( 1 + 5.80iT - 47T^{2} \)
53 \( 1 + (-3.06 - 1.76i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.95 + 3.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.908 - 1.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.0 - 6.39i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.290 + 0.503i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.75iT - 73T^{2} \)
79 \( 1 + (4.65 + 8.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.86 - 1.07i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.39 - 9.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.17iT - 97T^{2} \)
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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

−11.56447898433998703796543385450, −10.50510831761873143129098790282, −9.310787814919293078447563404470, −8.627839493741335117329746185302, −7.58611509227847646415879537139, −6.63939794608235170186425590176, −5.60433504187068422651312666665, −4.58435168223422397078829686595, −2.95016686189519207428471442057, −2.04070613049101012015052358194, 2.00762210868214363487569047403, 3.12299092614910901649427770538, 4.10676307667860299908768728579, 5.52728034949395419483585037175, 6.29886242258848461423397262550, 7.64845989050594571949432321053, 8.842885957550237548482697321899, 9.778712347858516895492511355252, 10.24414752034137288930832863010, 11.31125464043077024498938648004

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