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Properties

Label 2-507-13.12-c3-0-8
Degree $2$
Conductor $507$
Sign $0.999 + 0.0304i$
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  − 3.82i·2-s − 3·3-s − 6.63·4-s + 0.275i·5-s + 11.4i·6-s − 0.0981i·7-s − 5.20i·8-s + 9·9-s + 1.05·10-s + 0.749i·11-s + 19.9·12-s − 0.375·14-s − 0.826i·15-s − 73.0·16-s − 53.7·17-s − 34.4i·18-s + ⋯
L(s)  = 1  − 1.35i·2-s − 0.577·3-s − 0.829·4-s + 0.0246i·5-s + 0.781i·6-s − 0.00529i·7-s − 0.230i·8-s + 0.333·9-s + 0.0333·10-s + 0.0205i·11-s + 0.479·12-s − 0.00716·14-s − 0.0142i·15-s − 1.14·16-s − 0.767·17-s − 0.450i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.999 + 0.0304i$
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: \((2,\ 507,\ (\ :3/2),\ 0.999 + 0.0304i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9066601822\)
\(L(\frac12)\) \(\approx\) \(0.9066601822\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.82iT - 8T^{2} \)
5 \( 1 - 0.275iT - 125T^{2} \)
7 \( 1 + 0.0981iT - 343T^{2} \)
11 \( 1 - 0.749iT - 1.33e3T^{2} \)
17 \( 1 + 53.7T + 4.91e3T^{2} \)
19 \( 1 - 145. iT - 6.85e3T^{2} \)
23 \( 1 + 29.3T + 1.21e4T^{2} \)
29 \( 1 + 267.T + 2.43e4T^{2} \)
31 \( 1 + 51.5iT - 2.97e4T^{2} \)
37 \( 1 - 133. iT - 5.06e4T^{2} \)
41 \( 1 - 430. iT - 6.89e4T^{2} \)
43 \( 1 - 282.T + 7.95e4T^{2} \)
47 \( 1 - 212. iT - 1.03e5T^{2} \)
53 \( 1 - 573.T + 1.48e5T^{2} \)
59 \( 1 + 495. iT - 2.05e5T^{2} \)
61 \( 1 + 310.T + 2.26e5T^{2} \)
67 \( 1 + 103. iT - 3.00e5T^{2} \)
71 \( 1 + 203. iT - 3.57e5T^{2} \)
73 \( 1 - 685. iT - 3.89e5T^{2} \)
79 \( 1 - 636.T + 4.93e5T^{2} \)
83 \( 1 - 506. iT - 5.71e5T^{2} \)
89 \( 1 + 700. iT - 7.04e5T^{2} \)
97 \( 1 - 874. iT - 9.12e5T^{2} \)
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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

−10.69463148795398087487060517166, −9.908582172487096366612101427108, −9.123023254514827690220390662883, −7.87462629461939396909955619473, −6.73034533372013549314264398556, −5.73095373048630973741073324724, −4.44698623314551785627254552423, −3.57074472864558862894692653074, −2.24679839976769890968184627449, −1.13611915921552176665054512630, 0.33329725382282219916228210878, 2.33966773452288821168465041630, 4.17986837924461452884928975948, 5.17694679859191135403422137424, 5.89797653265359277950965504661, 7.02444971677619084278306706361, 7.29475962862547351985437689672, 8.727611577965261402089041203697, 9.172218850140396020506546112218, 10.68023331997296550882638800169

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