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Properties

Label 2-507-13.12-c3-0-63
Degree $2$
Conductor $507$
Sign $-0.246 + 0.969i$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.213i·2-s + 3·3-s + 7.95·4-s − 15.3i·5-s − 0.641i·6-s − 32.3i·7-s − 3.40i·8-s + 9·9-s − 3.27·10-s + 29.5i·11-s + 23.8·12-s − 6.92·14-s − 46.0i·15-s + 62.9·16-s + 78.1·17-s − 1.92i·18-s + ⋯
L(s)  = 1  − 0.0755i·2-s + 0.577·3-s + 0.994·4-s − 1.37i·5-s − 0.0436i·6-s − 1.74i·7-s − 0.150i·8-s + 0.333·9-s − 0.103·10-s + 0.811i·11-s + 0.574·12-s − 0.132·14-s − 0.792i·15-s + 0.982·16-s + 1.11·17-s − 0.0251i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.246 + 0.969i$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ -0.246 + 0.969i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.131814852\)
\(L(\frac12)\) \(\approx\) \(3.131814852\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
13 \( 1 \)
good2 \( 1 + 0.213iT - 8T^{2} \)
5 \( 1 + 15.3iT - 125T^{2} \)
7 \( 1 + 32.3iT - 343T^{2} \)
11 \( 1 - 29.5iT - 1.33e3T^{2} \)
17 \( 1 - 78.1T + 4.91e3T^{2} \)
19 \( 1 + 10.6iT - 6.85e3T^{2} \)
23 \( 1 - 26.8T + 1.21e4T^{2} \)
29 \( 1 + 190.T + 2.43e4T^{2} \)
31 \( 1 - 128. iT - 2.97e4T^{2} \)
37 \( 1 + 379. iT - 5.06e4T^{2} \)
41 \( 1 - 464. iT - 6.89e4T^{2} \)
43 \( 1 + 322.T + 7.95e4T^{2} \)
47 \( 1 + 248. iT - 1.03e5T^{2} \)
53 \( 1 - 740.T + 1.48e5T^{2} \)
59 \( 1 + 340. iT - 2.05e5T^{2} \)
61 \( 1 + 590.T + 2.26e5T^{2} \)
67 \( 1 - 340. iT - 3.00e5T^{2} \)
71 \( 1 - 36.2iT - 3.57e5T^{2} \)
73 \( 1 + 164. iT - 3.89e5T^{2} \)
79 \( 1 + 327.T + 4.93e5T^{2} \)
83 \( 1 - 1.40e3iT - 5.71e5T^{2} \)
89 \( 1 - 736. iT - 7.04e5T^{2} \)
97 \( 1 + 1.49e3iT - 9.12e5T^{2} \)
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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

−10.15178221757646046411263546029, −9.521276001329490253106380257912, −8.279577593221516172011529874484, −7.49172674875461745344717845012, −6.92413737695660088328538770714, −5.41825161910720841852414956141, −4.32977345620297206988022442034, −3.41457714311057084170979554366, −1.75212143429243122955631423563, −0.893783031856705138040717811662, 1.89800054516383205494305522840, 2.84755344176282426198086722100, 3.36554586629300034491551944174, 5.55866869136309174813721827002, 6.12200113108826086935502171270, 7.13464122026741034057005720212, 7.981378416412214022825870691194, 8.894915625957430099635896470371, 9.966318640252989286360398156927, 10.78783335469319271970918215036

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