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Properties

Label 2-507-13.12-c3-0-14
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  − 1.05i·2-s − 3·3-s + 6.89·4-s + 17.8i·5-s + 3.15i·6-s + 30.1i·7-s − 15.6i·8-s + 9·9-s + 18.8·10-s + 50.8i·11-s − 20.6·12-s + 31.7·14-s − 53.6i·15-s + 38.6·16-s + 2.99·17-s − 9.46i·18-s + ⋯
L(s)  = 1  − 0.371i·2-s − 0.577·3-s + 0.861·4-s + 1.60i·5-s + 0.214i·6-s + 1.63i·7-s − 0.692i·8-s + 0.333·9-s + 0.594·10-s + 1.39i·11-s − 0.497·12-s + 0.606·14-s − 0.923i·15-s + 0.604·16-s + 0.0426·17-s − 0.123i·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

\(L(2)\) \(\approx\) \(1.726637050\)
\(L(\frac12)\) \(\approx\) \(1.726637050\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.05iT - 8T^{2} \)
5 \( 1 - 17.8iT - 125T^{2} \)
7 \( 1 - 30.1iT - 343T^{2} \)
11 \( 1 - 50.8iT - 1.33e3T^{2} \)
17 \( 1 - 2.99T + 4.91e3T^{2} \)
19 \( 1 - 72.7iT - 6.85e3T^{2} \)
23 \( 1 - 41.9T + 1.21e4T^{2} \)
29 \( 1 + 135.T + 2.43e4T^{2} \)
31 \( 1 + 316. iT - 2.97e4T^{2} \)
37 \( 1 + 261. iT - 5.06e4T^{2} \)
41 \( 1 - 198. iT - 6.89e4T^{2} \)
43 \( 1 - 201.T + 7.95e4T^{2} \)
47 \( 1 + 97.3iT - 1.03e5T^{2} \)
53 \( 1 + 150.T + 1.48e5T^{2} \)
59 \( 1 + 497. iT - 2.05e5T^{2} \)
61 \( 1 - 525.T + 2.26e5T^{2} \)
67 \( 1 - 777. iT - 3.00e5T^{2} \)
71 \( 1 - 1.01e3iT - 3.57e5T^{2} \)
73 \( 1 + 612. iT - 3.89e5T^{2} \)
79 \( 1 - 718.T + 4.93e5T^{2} \)
83 \( 1 - 397. iT - 5.71e5T^{2} \)
89 \( 1 - 648. iT - 7.04e5T^{2} \)
97 \( 1 - 272. iT - 9.12e5T^{2} \)
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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

−11.07229523410791931792157373708, −10.04074660569062224384826960831, −9.513654319515680142211440259264, −7.82153713191242826778456917865, −7.06208384138856241517497180119, −6.22865016483585111156941672119, −5.54122348122814705224172863002, −3.82638277189863858307667854186, −2.53746572695935372436006323787, −2.01496570462266934776951620744, 0.57801041169481246713897488801, 1.35508676638821066039434631935, 3.41208361996586765925284427593, 4.65936454809193768373703491142, 5.45122025382654441759535899680, 6.50205874302515100634083187194, 7.36689059175280787447761323897, 8.257760699532425081833866987161, 9.137056362152279343872218173339, 10.45548660749975325079681528416

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