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Properties

Label 2-507-39.8-c1-0-39
Degree $2$
Conductor $507$
Sign $-0.957 + 0.289i$
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  + (1.69 − 1.69i)2-s − 1.73·3-s − 3.73i·4-s + (1.23 − 1.23i)5-s + (−2.93 + 2.93i)6-s + (−2.93 − 2.93i)8-s + 2.99·9-s − 4.19i·10-s + (−4.62 − 4.62i)11-s + 6.46i·12-s + (−2.14 + 2.14i)15-s − 2.46·16-s + (5.07 − 5.07i)18-s + (−4.62 − 4.62i)20-s − 15.6·22-s + ⋯
L(s)  = 1  + (1.19 − 1.19i)2-s − 1.00·3-s − 1.86i·4-s + (0.554 − 0.554i)5-s + (−1.19 + 1.19i)6-s + (−1.03 − 1.03i)8-s + 0.999·9-s − 1.32i·10-s + (−1.39 − 1.39i)11-s + 1.86i·12-s + (−0.554 + 0.554i)15-s − 0.616·16-s + (1.19 − 1.19i)18-s + (−1.03 − 1.03i)20-s − 3.33·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.957 + 0.289i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278942 - 1.88387i\)
\(L(\frac12)\) \(\approx\) \(0.278942 - 1.88387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
13 \( 1 \)
good2 \( 1 + (-1.69 + 1.69i)T - 2iT^{2} \)
5 \( 1 + (-1.23 + 1.23i)T - 5iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + (4.62 + 4.62i)T + 11iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31iT^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 + (-5.53 + 5.53i)T - 41iT^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (-7.10 - 7.10i)T + 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-0.332 - 0.332i)T + 59iT^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 + (-11.3 + 11.3i)T - 71iT^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + (-8.91 + 8.91i)T - 83iT^{2} \)
89 \( 1 + (-3.05 - 3.05i)T + 89iT^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−10.81019105266373859182213525130, −10.19236306257401698967495771334, −9.094820010645401307561129777415, −7.69921652065999922176079735262, −6.10944078666756845558280651270, −5.51470551455422474227637398169, −4.89524815111895920250920752605, −3.68145883692429373569907853483, −2.38214065357645810577182725926, −0.895613408524568201116100160247, 2.44387846737510216682798653360, 4.10684717872662527560332285157, 5.03069338811214912309796883350, 5.64167144969663204694464170131, 6.63017136723267024605730952332, 7.19960942562484064073154074333, 8.076241568783003249219061384957, 9.842969862528991093813045696606, 10.38994019678840381987569135138, 11.52937675932357181348205729471

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