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Properties

Label 2-1014-39.8-c1-0-12
Degree $2$
Conductor $1014$
Sign $0.724 - 0.689i$
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 + (−1.64 − 0.542i)3-s − 1.00i·4-s + (−2.32 + 2.32i)5-s + (1.54 − 0.779i)6-s + (1.76 − 1.76i)7-s + (0.707 + 0.707i)8-s + (2.41 + 1.78i)9-s − 3.28i·10-s + (−1.08 − 1.08i)11-s + (−0.542 + 1.64i)12-s + 2.49i·14-s + (5.08 − 2.56i)15-s − 1.00·16-s − 5.73·17-s + (−2.96 + 0.443i)18-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.949 − 0.313i)3-s − 0.500i·4-s + (−1.04 + 1.04i)5-s + (0.631 − 0.318i)6-s + (0.667 − 0.667i)7-s + (0.250 + 0.250i)8-s + (0.803 + 0.594i)9-s − 1.04i·10-s + (−0.327 − 0.327i)11-s + (−0.156 + 0.474i)12-s + 0.667i·14-s + (1.31 − 0.662i)15-s − 0.250·16-s − 1.39·17-s + (−0.699 + 0.104i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.724 - 0.689i$
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: \((2,\ 1014,\ (\ :1/2),\ 0.724 - 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6222931146\)
\(L(\frac12)\) \(\approx\) \(0.6222931146\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 0.542i)T \)
13 \( 1 \)
good5 \( 1 + (2.32 - 2.32i)T - 5iT^{2} \)
7 \( 1 + (-1.76 + 1.76i)T - 7iT^{2} \)
11 \( 1 + (1.08 + 1.08i)T + 11iT^{2} \)
17 \( 1 + 5.73T + 17T^{2} \)
19 \( 1 + (2.28 + 2.28i)T + 19iT^{2} \)
23 \( 1 - 4.65T + 23T^{2} \)
29 \( 1 + 4.65iT - 29T^{2} \)
31 \( 1 + (-3.82 - 3.82i)T + 31iT^{2} \)
37 \( 1 + (-3.05 + 3.05i)T - 37iT^{2} \)
41 \( 1 + (-0.410 + 0.410i)T - 41iT^{2} \)
43 \( 1 + 0.222iT - 43T^{2} \)
47 \( 1 + (-7.65 - 7.65i)T + 47iT^{2} \)
53 \( 1 - 10.9iT - 53T^{2} \)
59 \( 1 + (-4.65 - 4.65i)T + 59iT^{2} \)
61 \( 1 - 3.06T + 61T^{2} \)
67 \( 1 + (0.533 + 0.533i)T + 67iT^{2} \)
71 \( 1 + (-5.48 + 5.48i)T - 71iT^{2} \)
73 \( 1 + (2.28 - 2.28i)T - 73iT^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 + (1.39 - 1.39i)T - 83iT^{2} \)
89 \( 1 + (-6.41 - 6.41i)T + 89iT^{2} \)
97 \( 1 + (-11.5 - 11.5i)T + 97iT^{2} \)
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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

−10.47647764115248771027844081101, −9.120435172931464414112989951629, −8.057378649241305472139670835541, −7.43739620867742869421829162678, −6.84073211087736005073966541279, −6.11640059810714550782580678634, −4.81238092185740469536555170229, −4.14129575218656219327784722429, −2.50066798466530458330036160694, −0.74487313834773667351328543400, 0.63139704764331052689136678275, 2.05859881093099471785705833495, 3.78122224342001892524053992383, 4.66876599907633466873957502467, 5.16965047482440729033766571559, 6.51149085482550677922194827377, 7.51726546724398134338338310730, 8.494293651963715791517752651493, 8.883508982032539373540288661449, 9.938435007392189878083961591838

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