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Properties

Label 2-1014-13.8-c2-0-3
Degree $2$
Conductor $1014$
Sign $0.471 - 0.881i$
Analytic cond. $27.6294$
Root an. cond. $5.25637$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + 1.73·3-s + 2i·4-s + (−5.04 − 5.04i)5-s + (−1.73 − 1.73i)6-s + (−3.68 + 3.68i)7-s + (2 − 2i)8-s + 2.99·9-s + 10.0i·10-s + (5.36 − 5.36i)11-s + 3.46i·12-s + 7.36·14-s + (−8.74 − 8.74i)15-s − 4·16-s + 16.1i·17-s + (−2.99 − 2.99i)18-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 0.577·3-s + 0.5i·4-s + (−1.00 − 1.00i)5-s + (−0.288 − 0.288i)6-s + (−0.525 + 0.525i)7-s + (0.250 − 0.250i)8-s + 0.333·9-s + 1.00i·10-s + (0.487 − 0.487i)11-s + 0.288i·12-s + 0.525·14-s + (−0.582 − 0.582i)15-s − 0.250·16-s + 0.950i·17-s + (−0.166 − 0.166i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(27.6294\)
Root analytic conductor: \(5.25637\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (775, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1),\ 0.471 - 0.881i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6317974706\)
\(L(\frac12)\) \(\approx\) \(0.6317974706\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 - 1.73T \)
13 \( 1 \)
good5 \( 1 + (5.04 + 5.04i)T + 25iT^{2} \)
7 \( 1 + (3.68 - 3.68i)T - 49iT^{2} \)
11 \( 1 + (-5.36 + 5.36i)T - 121iT^{2} \)
17 \( 1 - 16.1iT - 289T^{2} \)
19 \( 1 + (7.23 + 7.23i)T + 361iT^{2} \)
23 \( 1 + 9.57iT - 529T^{2} \)
29 \( 1 + 33.1T + 841T^{2} \)
31 \( 1 + (34.2 + 34.2i)T + 961iT^{2} \)
37 \( 1 + (46.2 - 46.2i)T - 1.36e3iT^{2} \)
41 \( 1 + (-38.5 - 38.5i)T + 1.68e3iT^{2} \)
43 \( 1 - 47.0iT - 1.84e3T^{2} \)
47 \( 1 + (-47.8 + 47.8i)T - 2.20e3iT^{2} \)
53 \( 1 - 67.5T + 2.80e3T^{2} \)
59 \( 1 + (53.2 - 53.2i)T - 3.48e3iT^{2} \)
61 \( 1 - 70.5T + 3.72e3T^{2} \)
67 \( 1 + (-30.5 - 30.5i)T + 4.48e3iT^{2} \)
71 \( 1 + (-78.1 - 78.1i)T + 5.04e3iT^{2} \)
73 \( 1 + (36.8 - 36.8i)T - 5.32e3iT^{2} \)
79 \( 1 + 13.3T + 6.24e3T^{2} \)
83 \( 1 + (-93.1 - 93.1i)T + 6.88e3iT^{2} \)
89 \( 1 + (35.5 - 35.5i)T - 7.92e3iT^{2} \)
97 \( 1 + (-38.4 - 38.4i)T + 9.40e3iT^{2} \)
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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

−9.630634527241523685936443877115, −8.986563165289521694809268753687, −8.442474051557083318120143666752, −7.79971398950023685705991156159, −6.71834567244662343823114669105, −5.54035895894901492463652981640, −4.19614764496762533234324570395, −3.68818700118920164846850819470, −2.44158506841706289462841719478, −1.09409815613968447112081690025, 0.24777311558598340690355410448, 2.05886881419050119101673191258, 3.45747057898898725271240350110, 3.97243740618665428759353691528, 5.40599580496101870604867365489, 6.71540776708335639602554727147, 7.28232935862892293175980358872, 7.60944269771140663205387506852, 8.876254010085205197567806259122, 9.409281709184104167697045360731

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