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Properties

Label 2-1014-39.8-c1-0-7
Degree $2$
Conductor $1014$
Sign $0.315 - 0.949i$
Analytic cond. $8.09683$
Root an. cond. $2.84549$
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  + (0.707 − 0.707i)2-s + (−1.53 + 0.796i)3-s − 1.00i·4-s + (−0.428 + 0.428i)5-s + (−0.524 + 1.65i)6-s + (−0.538 + 0.538i)7-s + (−0.707 − 0.707i)8-s + (1.73 − 2.44i)9-s + 0.606i·10-s + (2.97 + 2.97i)11-s + (0.796 + 1.53i)12-s + 0.761i·14-s + (0.317 − 1.00i)15-s − 1.00·16-s − 5.24·17-s + (−0.507 − 2.95i)18-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.888 + 0.459i)3-s − 0.500i·4-s + (−0.191 + 0.191i)5-s + (−0.214 + 0.673i)6-s + (−0.203 + 0.203i)7-s + (−0.250 − 0.250i)8-s + (0.577 − 0.816i)9-s + 0.191i·10-s + (0.895 + 0.895i)11-s + (0.229 + 0.444i)12-s + 0.203i·14-s + (0.0820 − 0.258i)15-s − 0.250·16-s − 1.27·17-s + (−0.119 − 0.696i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.315 - 0.949i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.315 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.093291159\)
\(L(\frac12)\) \(\approx\) \(1.093291159\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (1.53 - 0.796i)T \)
13 \( 1 \)
good5 \( 1 + (0.428 - 0.428i)T - 5iT^{2} \)
7 \( 1 + (0.538 - 0.538i)T - 7iT^{2} \)
11 \( 1 + (-2.97 - 2.97i)T + 11iT^{2} \)
17 \( 1 + 5.24T + 17T^{2} \)
19 \( 1 + (2.41 + 2.41i)T + 19iT^{2} \)
23 \( 1 - 1.86T + 23T^{2} \)
29 \( 1 - 8.70iT - 29T^{2} \)
31 \( 1 + (-2.68 - 2.68i)T + 31iT^{2} \)
37 \( 1 + (4.15 - 4.15i)T - 37iT^{2} \)
41 \( 1 + (-6.27 + 6.27i)T - 41iT^{2} \)
43 \( 1 - 1.95iT - 43T^{2} \)
47 \( 1 + (-5.73 - 5.73i)T + 47iT^{2} \)
53 \( 1 - 9.01iT - 53T^{2} \)
59 \( 1 + (-6.10 - 6.10i)T + 59iT^{2} \)
61 \( 1 + 8.13T + 61T^{2} \)
67 \( 1 + (0.0740 + 0.0740i)T + 67iT^{2} \)
71 \( 1 + (-7.37 + 7.37i)T - 71iT^{2} \)
73 \( 1 + (5.57 - 5.57i)T - 73iT^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + (-0.996 + 0.996i)T - 83iT^{2} \)
89 \( 1 + (-4.62 - 4.62i)T + 89iT^{2} \)
97 \( 1 + (11.1 + 11.1i)T + 97iT^{2} \)
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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

−10.47829142352338999770368074223, −9.256637774093731181324418033718, −9.009310959634639789963493304571, −7.13757098939823428275013742704, −6.70094903949023111482599406311, −5.70595635209322293620416881060, −4.67371948432120678007788324755, −4.14663914929796855091420199993, −2.92987703145888283818634239071, −1.43129228507840089909841717196, 0.49540309755127825585421892244, 2.23516137012976657834445795599, 3.88314280050616699254707315052, 4.51862257920285506509836132374, 5.71150708937496237325612019203, 6.36098419072730141385598222788, 6.93614241510307684181020479133, 8.038361690264475752728988084366, 8.692921038444340120894998055831, 9.853752738341965946920500316418

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