Properties

Label 2-507-13.12-c3-0-54
Degree $2$
Conductor $507$
Sign $0.691 + 0.722i$
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  − 0.447i·2-s + 3·3-s + 7.79·4-s − 1.93i·5-s − 1.34i·6-s − 8.14i·7-s − 7.06i·8-s + 9·9-s − 0.863·10-s + 8.40i·11-s + 23.3·12-s − 3.64·14-s − 5.79i·15-s + 59.2·16-s + 52.1·17-s − 4.02i·18-s + ⋯
L(s)  = 1  − 0.158i·2-s + 0.577·3-s + 0.974·4-s − 0.172i·5-s − 0.0913i·6-s − 0.439i·7-s − 0.312i·8-s + 0.333·9-s − 0.0273·10-s + 0.230i·11-s + 0.562·12-s − 0.0695·14-s − 0.0997i·15-s + 0.925·16-s + 0.743·17-s − 0.0527i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.691 + 0.722i$
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: \((2,\ 507,\ (\ :3/2),\ 0.691 + 0.722i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.240874996\)
\(L(\frac12)\) \(\approx\) \(3.240874996\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.447iT - 8T^{2} \)
5 \( 1 + 1.93iT - 125T^{2} \)
7 \( 1 + 8.14iT - 343T^{2} \)
11 \( 1 - 8.40iT - 1.33e3T^{2} \)
17 \( 1 - 52.1T + 4.91e3T^{2} \)
19 \( 1 + 48.8iT - 6.85e3T^{2} \)
23 \( 1 + 88.9T + 1.21e4T^{2} \)
29 \( 1 - 191.T + 2.43e4T^{2} \)
31 \( 1 + 115. iT - 2.97e4T^{2} \)
37 \( 1 + 136. iT - 5.06e4T^{2} \)
41 \( 1 + 436. iT - 6.89e4T^{2} \)
43 \( 1 + 202.T + 7.95e4T^{2} \)
47 \( 1 - 618. iT - 1.03e5T^{2} \)
53 \( 1 + 453.T + 1.48e5T^{2} \)
59 \( 1 - 500. iT - 2.05e5T^{2} \)
61 \( 1 - 480.T + 2.26e5T^{2} \)
67 \( 1 + 886. iT - 3.00e5T^{2} \)
71 \( 1 + 123. iT - 3.57e5T^{2} \)
73 \( 1 - 673. iT - 3.89e5T^{2} \)
79 \( 1 + 681.T + 4.93e5T^{2} \)
83 \( 1 - 939. iT - 5.71e5T^{2} \)
89 \( 1 + 754. iT - 7.04e5T^{2} \)
97 \( 1 + 1.05e3iT - 9.12e5T^{2} \)
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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

−10.39448279057745934615096749970, −9.642156598697942268310897778651, −8.518929200914575191548328041598, −7.59755345954454586863045201761, −6.90870038983612159458290771024, −5.83881399332433986440044134003, −4.47702838811519444542322121695, −3.31168366818523716889266148030, −2.29588585372856779077948357696, −1.01016903099755044688322523601, 1.41216085686809790400890131062, 2.64192912302944430821923215073, 3.48257660048144029752868176110, 5.05735666995576178684938395547, 6.17368758724613128628801988606, 6.93214956014144487621333714040, 8.015964954963011922534720222193, 8.552444908810295407311527036951, 9.911291864194382133258342419444, 10.44998879422258794288555829698

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