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Properties

Label 2-1014-39.8-c1-0-31
Degree $2$
Conductor $1014$
Sign $0.956 - 0.290i$
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 + (2.02 − 2.02i)5-s + (−1.65 + 0.524i)6-s + (2.53 − 2.53i)7-s + (0.707 + 0.707i)8-s + (1.73 + 2.44i)9-s + 2.85i·10-s + (2.97 + 2.97i)11-s + (0.796 − 1.53i)12-s + 3.58i·14-s + (4.71 − 1.49i)15-s − 1.00·16-s − 3.45·17-s + (−2.95 − 0.507i)18-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.888 + 0.459i)3-s − 0.500i·4-s + (0.903 − 0.903i)5-s + (−0.673 + 0.214i)6-s + (0.959 − 0.959i)7-s + (0.250 + 0.250i)8-s + (0.577 + 0.816i)9-s + 0.903i·10-s + (0.895 + 0.895i)11-s + (0.229 − 0.444i)12-s + 0.959i·14-s + (1.21 − 0.387i)15-s − 0.250·16-s − 0.837·17-s + (−0.696 − 0.119i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.956 - 0.290i$
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: \((2,\ 1014,\ (\ :1/2),\ 0.956 - 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.251064563\)
\(L(\frac12)\) \(\approx\) \(2.251064563\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 + 2.02i)T - 5iT^{2} \)
7 \( 1 + (-2.53 + 2.53i)T - 7iT^{2} \)
11 \( 1 + (-2.97 - 2.97i)T + 11iT^{2} \)
17 \( 1 + 3.45T + 17T^{2} \)
19 \( 1 + (1.58 + 1.58i)T + 19iT^{2} \)
23 \( 1 + 3.03T + 23T^{2} \)
29 \( 1 - 2.01iT - 29T^{2} \)
31 \( 1 + (1.21 + 1.21i)T + 31iT^{2} \)
37 \( 1 + (-3.42 + 3.42i)T - 37iT^{2} \)
41 \( 1 + (-5.61 + 5.61i)T - 41iT^{2} \)
43 \( 1 + 1.95iT - 43T^{2} \)
47 \( 1 + (0.957 + 0.957i)T + 47iT^{2} \)
53 \( 1 - 7.22iT - 53T^{2} \)
59 \( 1 + (7.27 + 7.27i)T + 59iT^{2} \)
61 \( 1 - 0.274T + 61T^{2} \)
67 \( 1 + (-3.00 - 3.00i)T + 67iT^{2} \)
71 \( 1 + (7.80 - 7.80i)T - 71iT^{2} \)
73 \( 1 + (10.0 - 10.0i)T - 73iT^{2} \)
79 \( 1 - 1.58T + 79T^{2} \)
83 \( 1 + (2.58 - 2.58i)T - 83iT^{2} \)
89 \( 1 + (6.96 + 6.96i)T + 89iT^{2} \)
97 \( 1 + (-5.67 - 5.67i)T + 97iT^{2} \)
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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

−9.704584447323801531011459449674, −9.124286271121385477961207910264, −8.528680156361251784226673455968, −7.59041785681794270392148368861, −6.90654454756523040012301755783, −5.63352355327066847319637880748, −4.54946830545572831101529610977, −4.18653491738875882177648844134, −2.16190075243257430093341138548, −1.36512432310787874508685886184, 1.55028457427433649377913096987, 2.30649655488144933651730123545, 3.14668229433337335186434899990, 4.37858128375726621188898680145, 6.00384368866945115244374236684, 6.48638074277871667625390554672, 7.68227567398293203035165164560, 8.450495065497752484961185456670, 9.043681522669818481844001457825, 9.752349037374283332183452682053

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