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Properties

Label 2-1014-39.8-c1-0-45
Degree $2$
Conductor $1014$
Sign $-0.0667 + 0.997i$
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 + (1.72 − 0.129i)3-s − 1.00i·4-s + (1.54 − 1.54i)5-s + (1.12 − 1.31i)6-s + (0.651 − 0.651i)7-s + (−0.707 − 0.707i)8-s + (2.96 − 0.448i)9-s − 2.18i·10-s + (−2.32 − 2.32i)11-s + (−0.129 − 1.72i)12-s − 0.921i·14-s + (2.46 − 2.86i)15-s − 1.00·16-s − 7.60·17-s + (1.78 − 2.41i)18-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.997 − 0.0750i)3-s − 0.500i·4-s + (0.691 − 0.691i)5-s + (0.461 − 0.536i)6-s + (0.246 − 0.246i)7-s + (−0.250 − 0.250i)8-s + (0.988 − 0.149i)9-s − 0.691i·10-s + (−0.700 − 0.700i)11-s + (−0.0375 − 0.498i)12-s − 0.246i·14-s + (0.637 − 0.740i)15-s − 0.250·16-s − 1.84·17-s + (0.419 − 0.569i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0667 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0667 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.0667 + 0.997i$
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: \((2,\ 1014,\ (\ :1/2),\ -0.0667 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.176176593\)
\(L(\frac12)\) \(\approx\) \(3.176176593\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.129i)T \)
13 \( 1 \)
good5 \( 1 + (-1.54 + 1.54i)T - 5iT^{2} \)
7 \( 1 + (-0.651 + 0.651i)T - 7iT^{2} \)
11 \( 1 + (2.32 + 2.32i)T + 11iT^{2} \)
17 \( 1 + 7.60T + 17T^{2} \)
19 \( 1 + (2.61 + 2.61i)T + 19iT^{2} \)
23 \( 1 - 6.68T + 23T^{2} \)
29 \( 1 - 7.08iT - 29T^{2} \)
31 \( 1 + (-6.55 - 6.55i)T + 31iT^{2} \)
37 \( 1 + (-1.13 + 1.13i)T - 37iT^{2} \)
41 \( 1 + (-5.29 + 5.29i)T - 41iT^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 + (-1.78 - 1.78i)T + 47iT^{2} \)
53 \( 1 + 7.08iT - 53T^{2} \)
59 \( 1 + (4.95 + 4.95i)T + 59iT^{2} \)
61 \( 1 - 2.47T + 61T^{2} \)
67 \( 1 + (1.79 + 1.79i)T + 67iT^{2} \)
71 \( 1 + (4.60 - 4.60i)T - 71iT^{2} \)
73 \( 1 + (-7.82 + 7.82i)T - 73iT^{2} \)
79 \( 1 - 2.90T + 79T^{2} \)
83 \( 1 + (4.21 - 4.21i)T - 83iT^{2} \)
89 \( 1 + (-1.20 - 1.20i)T + 89iT^{2} \)
97 \( 1 + (-1.61 - 1.61i)T + 97iT^{2} \)
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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

−9.583782386316445566283975284397, −8.916953488861036664537294454700, −8.441106662573908970536836532510, −7.16772001046892087730475072127, −6.33392964382403804639626191986, −5.00370447821994168320026259028, −4.54036942305602925558097464495, −3.18430489133832287116002499122, −2.35159682690209575485751154225, −1.17944917064118927182806812284, 2.25744176076166588802346813157, 2.59474500412481103943884020787, 4.09810372384065063014934717858, 4.78279436193035607378121427048, 6.06751709073985617373169201997, 6.79310100922212921845068265922, 7.61945174580373499670101298802, 8.443093391511250556981134231381, 9.225567407708758510797959480167, 10.10224482161800240497069559645

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