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Properties

Label 2-1014-39.8-c1-0-36
Degree $2$
Conductor $1014$
Sign $0.635 + 0.772i$
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 + (0.796 + 1.53i)3-s − 1.00i·4-s + (2.76 − 2.76i)5-s + (1.65 + 0.524i)6-s + (1.79 − 1.79i)7-s + (−0.707 − 0.707i)8-s + (−1.73 + 2.44i)9-s − 3.90i·10-s + (0.412 + 0.412i)11-s + (1.53 − 0.796i)12-s − 2.54i·14-s + (6.44 + 2.04i)15-s − 1.00·16-s − 1.09·17-s + (0.507 + 2.95i)18-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.459 + 0.888i)3-s − 0.500i·4-s + (1.23 − 1.23i)5-s + (0.673 + 0.214i)6-s + (0.678 − 0.678i)7-s + (−0.250 − 0.250i)8-s + (−0.577 + 0.816i)9-s − 1.23i·10-s + (0.124 + 0.124i)11-s + (0.444 − 0.229i)12-s − 0.678i·14-s + (1.66 + 0.529i)15-s − 0.250·16-s − 0.265·17-s + (0.119 + 0.696i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.635 + 0.772i$
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: \((2,\ 1014,\ (\ :1/2),\ 0.635 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.090507089\)
\(L(\frac12)\) \(\approx\) \(3.090507089\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.796 - 1.53i)T \)
13 \( 1 \)
good5 \( 1 + (-2.76 + 2.76i)T - 5iT^{2} \)
7 \( 1 + (-1.79 + 1.79i)T - 7iT^{2} \)
11 \( 1 + (-0.412 - 0.412i)T + 11iT^{2} \)
17 \( 1 + 1.09T + 17T^{2} \)
19 \( 1 + (-0.971 - 0.971i)T + 19iT^{2} \)
23 \( 1 - 1.75T + 23T^{2} \)
29 \( 1 - 5.92iT - 29T^{2} \)
31 \( 1 + (6.49 + 6.49i)T + 31iT^{2} \)
37 \( 1 + (2.18 - 2.18i)T - 37iT^{2} \)
41 \( 1 + (3.74 - 3.74i)T - 41iT^{2} \)
43 \( 1 + 3.76iT - 43T^{2} \)
47 \( 1 + (-5.51 - 5.51i)T + 47iT^{2} \)
53 \( 1 - 3.04iT - 53T^{2} \)
59 \( 1 + (-5.99 - 5.99i)T + 59iT^{2} \)
61 \( 1 - 9.34T + 61T^{2} \)
67 \( 1 + (4.66 + 4.66i)T + 67iT^{2} \)
71 \( 1 + (0.601 - 0.601i)T - 71iT^{2} \)
73 \( 1 + (-5.18 + 5.18i)T - 73iT^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + (-5.15 + 5.15i)T - 83iT^{2} \)
89 \( 1 + (-6.85 - 6.85i)T + 89iT^{2} \)
97 \( 1 + (0.433 + 0.433i)T + 97iT^{2} \)
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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

−9.861679631747554901813548063454, −9.150328188507540813988372801039, −8.583718715089085139262821333287, −7.43629634373018725354656172816, −5.98385275646538021071135311581, −5.17050707782317414499275863794, −4.63479884963688965296631583505, −3.71848060525703603812305294852, −2.31012103347525197637837438426, −1.31089078456558644923483431063, 1.86532960968696613001524439692, 2.57009363913508828878442426613, 3.57685894461616979849603425761, 5.28444193289211736671439062289, 5.86288764999891016376237392280, 6.79407345616072274538291144407, 7.20760910289814334907080021607, 8.358248966563161376603699998470, 9.026713615199661491721148244285, 9.976491618351339217021053109240

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