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Properties

Label 2-507-13.12-c3-0-9
Degree $2$
Conductor $507$
Sign $0.277 - 0.960i$
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  − 3.27i·2-s + 3·3-s − 2.75·4-s + 17.5i·5-s − 9.83i·6-s + 26.6i·7-s − 17.2i·8-s + 9·9-s + 57.5·10-s + 21.4i·11-s − 8.25·12-s + 87.5·14-s + 52.6i·15-s − 78.4·16-s − 83.9·17-s − 29.5i·18-s + ⋯
L(s)  = 1  − 1.15i·2-s + 0.577·3-s − 0.343·4-s + 1.56i·5-s − 0.669i·6-s + 1.44i·7-s − 0.760i·8-s + 0.333·9-s + 1.81·10-s + 0.586i·11-s − 0.198·12-s + 1.67·14-s + 0.905i·15-s − 1.22·16-s − 1.19·17-s − 0.386i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.277 - 0.960i$
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: \((2,\ 507,\ (\ :3/2),\ 0.277 - 0.960i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.712384406\)
\(L(\frac12)\) \(\approx\) \(1.712384406\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.27iT - 8T^{2} \)
5 \( 1 - 17.5iT - 125T^{2} \)
7 \( 1 - 26.6iT - 343T^{2} \)
11 \( 1 - 21.4iT - 1.33e3T^{2} \)
17 \( 1 + 83.9T + 4.91e3T^{2} \)
19 \( 1 + 77.1iT - 6.85e3T^{2} \)
23 \( 1 + 142.T + 1.21e4T^{2} \)
29 \( 1 - 134.T + 2.43e4T^{2} \)
31 \( 1 - 122. iT - 2.97e4T^{2} \)
37 \( 1 - 222. iT - 5.06e4T^{2} \)
41 \( 1 - 198. iT - 6.89e4T^{2} \)
43 \( 1 + 154.T + 7.95e4T^{2} \)
47 \( 1 - 78.7iT - 1.03e5T^{2} \)
53 \( 1 + 477.T + 1.48e5T^{2} \)
59 \( 1 - 42.9iT - 2.05e5T^{2} \)
61 \( 1 - 496.T + 2.26e5T^{2} \)
67 \( 1 + 484. iT - 3.00e5T^{2} \)
71 \( 1 - 382. iT - 3.57e5T^{2} \)
73 \( 1 + 193. iT - 3.89e5T^{2} \)
79 \( 1 - 1.04e3T + 4.93e5T^{2} \)
83 \( 1 - 861. iT - 5.71e5T^{2} \)
89 \( 1 + 967. iT - 7.04e5T^{2} \)
97 \( 1 - 591. iT - 9.12e5T^{2} \)
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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

−10.74138528461417777661250308676, −9.941252203018203674407225792155, −9.231939584787774440234655373698, −8.168283242630943070929154436088, −6.85372255985493746899547221048, −6.36575160605902505434325161100, −4.63686138769763964165089873191, −3.30475374416666824209388845007, −2.56807120266969172090784521421, −1.99381653647631680212680098514, 0.44557143937258746431278682704, 1.88600227558802377525365619761, 3.90147729129743520419596291194, 4.61575863197694879222361694809, 5.74282163714315301473258480436, 6.74033543456071486221633976129, 7.79098861009351468110164953125, 8.269294577982124641851318814241, 9.042971879533114740996980759139, 10.09182990905844807594996568321

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