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Properties

Label 2-370-185.142-c1-0-16
Degree $2$
Conductor $370$
Sign $-0.608 + 0.793i$
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  − i·2-s + (1.28 + 1.28i)3-s − 4-s + (−1.52 − 1.63i)5-s + (1.28 − 1.28i)6-s + (−3.01 − 3.01i)7-s + i·8-s + 0.323i·9-s + (−1.63 + 1.52i)10-s − 2.38i·11-s + (−1.28 − 1.28i)12-s − 1.44i·13-s + (−3.01 + 3.01i)14-s + (0.148 − 4.07i)15-s + 16-s − 2.09·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.744 + 0.744i)3-s − 0.5·4-s + (−0.680 − 0.732i)5-s + (0.526 − 0.526i)6-s + (−1.13 − 1.13i)7-s + 0.353i·8-s + 0.107i·9-s + (−0.517 + 0.481i)10-s − 0.720i·11-s + (−0.372 − 0.372i)12-s − 0.401i·13-s + (−0.805 + 0.805i)14-s + (0.0384 − 1.05i)15-s + 0.250·16-s − 0.508·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.608 + 0.793i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.468185 - 0.949586i\)
\(L(\frac12)\) \(\approx\) \(0.468185 - 0.949586i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (1.52 + 1.63i)T \)
37 \( 1 + (-5.78 + 1.89i)T \)
good3 \( 1 + (-1.28 - 1.28i)T + 3iT^{2} \)
7 \( 1 + (3.01 + 3.01i)T + 7iT^{2} \)
11 \( 1 + 2.38iT - 11T^{2} \)
13 \( 1 + 1.44iT - 13T^{2} \)
17 \( 1 + 2.09T + 17T^{2} \)
19 \( 1 + (-2.46 - 2.46i)T + 19iT^{2} \)
23 \( 1 - 0.168iT - 23T^{2} \)
29 \( 1 + (-4.74 + 4.74i)T - 29iT^{2} \)
31 \( 1 + (3.34 + 3.34i)T + 31iT^{2} \)
41 \( 1 - 6.22iT - 41T^{2} \)
43 \( 1 - 3.07iT - 43T^{2} \)
47 \( 1 + (4.67 + 4.67i)T + 47iT^{2} \)
53 \( 1 + (2.87 - 2.87i)T - 53iT^{2} \)
59 \( 1 + (6.78 + 6.78i)T + 59iT^{2} \)
61 \( 1 + (-6.94 - 6.94i)T + 61iT^{2} \)
67 \( 1 + (7.89 - 7.89i)T - 67iT^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + (4.83 + 4.83i)T + 73iT^{2} \)
79 \( 1 + (-5.31 - 5.31i)T + 79iT^{2} \)
83 \( 1 + (-8.43 + 8.43i)T - 83iT^{2} \)
89 \( 1 + (-11.6 + 11.6i)T - 89iT^{2} \)
97 \( 1 - 5.04T + 97T^{2} \)
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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

−10.97459016117619825958346417655, −9.974705166588812528095575293983, −9.476337347035813299505975803493, −8.503974128535003303860452607627, −7.65053526231189085610264426803, −6.17344480015919416547888329364, −4.56779554770329555793837214673, −3.76541255217649313483027285745, −3.08913277000028064983233499181, −0.65970699845296022466153006354, 2.39200391937552066787820599003, 3.37389583186678846262484747971, 4.95195928230176212378277592921, 6.44018272900372272479467031642, 6.95382731209845779067914614607, 7.84676954569861682445869282050, 8.822967351019412217748400450262, 9.505566304818643701015583896046, 10.76688604934987798525703982693, 12.06651671682236867456726115660

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