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Properties

Label 2-507-13.9-c1-0-7
Degree $2$
Conductor $507$
Sign $-0.0128 - 0.999i$
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  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (0.500 + 0.866i)4-s + 2·5-s + (0.499 + 0.866i)6-s + (2 + 3.46i)7-s − 3·8-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)10-s + (−2 + 3.46i)11-s + 12-s − 3.99·14-s + (1 − 1.73i)15-s + (0.500 − 0.866i)16-s + (−1 − 1.73i)17-s + 0.999·18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (0.250 + 0.433i)4-s + 0.894·5-s + (0.204 + 0.353i)6-s + (0.755 + 1.30i)7-s − 1.06·8-s + (−0.166 − 0.288i)9-s + (−0.316 + 0.547i)10-s + (−0.603 + 1.04i)11-s + 0.288·12-s − 1.06·14-s + (0.258 − 0.447i)15-s + (0.125 − 0.216i)16-s + (−0.242 − 0.420i)17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.0128 - 0.999i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11602 + 1.13042i\)
\(L(\frac12)\) \(\approx\) \(1.11602 + 1.13042i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6 - 10.3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.35278456111505393140591701449, −9.860223230300309532973443219191, −9.229718501172201749401543322655, −8.217346709228614328483705371552, −7.74811839956405163361973355805, −6.54800514714370983224343240132, −5.85732787959976694710321777273, −4.73897667355618378839530485033, −2.69806799877585015634573298336, −2.09495784516660473180817309851, 1.07960758519514121756710090243, 2.38957587810004361897357873419, 3.65206607630479313729802300122, 5.01380420270862060626962663481, 5.91110512932443020117134210164, 7.04714025670043888614968497425, 8.304382427255966794059718836807, 9.057260385329864963195085802075, 10.20407440138458392178190964046, 10.52099177368355250117715276635

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