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Properties

Label 2-507-39.5-c1-0-13
Degree $2$
Conductor $507$
Sign $-0.646 - 0.762i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.38 + 1.38i)2-s + (0.526 + 1.65i)3-s + 1.83i·4-s + (−1.04 − 1.04i)5-s + (−1.55 + 3.01i)6-s + (3.17 + 3.17i)7-s + (0.233 − 0.233i)8-s + (−2.44 + 1.73i)9-s − 2.89i·10-s + (−0.108 + 0.108i)11-s + (−3.02 + 0.964i)12-s + 8.77i·14-s + (1.17 − 2.27i)15-s + 4.30·16-s − 3.16·17-s + (−5.78 − 0.980i)18-s + ⋯
L(s)  = 1  + (0.978 + 0.978i)2-s + (0.303 + 0.952i)3-s + 0.915i·4-s + (−0.468 − 0.468i)5-s + (−0.634 + 1.22i)6-s + (1.19 + 1.19i)7-s + (0.0825 − 0.0825i)8-s + (−0.815 + 0.579i)9-s − 0.916i·10-s + (−0.0326 + 0.0326i)11-s + (−0.872 + 0.278i)12-s + 2.34i·14-s + (0.303 − 0.588i)15-s + 1.07·16-s − 0.768·17-s + (−1.36 − 0.231i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 - 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.646 - 0.762i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.646 - 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09734 + 2.36805i\)
\(L(\frac12)\) \(\approx\) \(1.09734 + 2.36805i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.526 - 1.65i)T \)
13 \( 1 \)
good2 \( 1 + (-1.38 - 1.38i)T + 2iT^{2} \)
5 \( 1 + (1.04 + 1.04i)T + 5iT^{2} \)
7 \( 1 + (-3.17 - 3.17i)T + 7iT^{2} \)
11 \( 1 + (0.108 - 0.108i)T - 11iT^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 + (-0.846 + 0.846i)T - 19iT^{2} \)
23 \( 1 + 6.70T + 23T^{2} \)
29 \( 1 + 1.98iT - 29T^{2} \)
31 \( 1 + (-3.64 + 3.64i)T - 31iT^{2} \)
37 \( 1 + (-2.31 - 2.31i)T + 37iT^{2} \)
41 \( 1 + (-5.91 - 5.91i)T + 41iT^{2} \)
43 \( 1 + 2.78iT - 43T^{2} \)
47 \( 1 + (-4.06 + 4.06i)T - 47iT^{2} \)
53 \( 1 + 0.628iT - 53T^{2} \)
59 \( 1 + (-6.20 + 6.20i)T - 59iT^{2} \)
61 \( 1 - 5.83T + 61T^{2} \)
67 \( 1 + (-0.475 + 0.475i)T - 67iT^{2} \)
71 \( 1 + (8.09 + 8.09i)T + 71iT^{2} \)
73 \( 1 + (3.92 + 3.92i)T + 73iT^{2} \)
79 \( 1 + 5.46T + 79T^{2} \)
83 \( 1 + (-6.40 - 6.40i)T + 83iT^{2} \)
89 \( 1 + (3.75 - 3.75i)T - 89iT^{2} \)
97 \( 1 + (-3.16 + 3.16i)T - 97iT^{2} \)
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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

−11.51233432957419796318931991514, −10.32572241245247022085098911922, −9.230800287212392882020385070703, −8.250188292937708686534617936451, −7.88492238242935281830405537814, −6.26136538220239006847518827809, −5.42495514159148431779484867896, −4.64238723237691846147702501604, −4.05273022018936642940605121806, −2.39527115625649938015953130926, 1.30797324033310513473844276558, 2.45443530582721040145938810690, 3.70759820614679356064810265062, 4.44667783288603479206786375518, 5.74552572267634420914803037032, 7.13054159286935095556904459267, 7.70468837275743556263856871206, 8.567010681227772831832160229049, 10.18538873842521645044067414236, 11.05315003504678222941255490247

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