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Properties

Label 2-507-13.3-c1-0-17
Degree $2$
Conductor $507$
Sign $0.872 - 0.488i$
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 + 5-s + (−0.499 + 0.866i)6-s + (1 − 1.73i)7-s + 3·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−1 − 1.73i)11-s + 12-s + 1.99·14-s + (0.5 + 0.866i)15-s + (0.500 + 0.866i)16-s + (3.5 − 6.06i)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 + 0.447·5-s + (−0.204 + 0.353i)6-s + (0.377 − 0.654i)7-s + 1.06·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (−0.301 − 0.522i)11-s + 0.288·12-s + 0.534·14-s + (0.129 + 0.223i)15-s + (0.125 + 0.216i)16-s + (0.848 − 1.47i)17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22350 + 0.580530i\)
\(L(\frac12)\) \(\approx\) \(2.22350 + 0.580530i\)
\(L(\frac{3}{2})\) 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 \( 1 - T + 5T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.90087348568912667803778320753, −10.02273523098508057396530509239, −9.434663920059725660171715396629, −7.984221796721787185930510575880, −7.43575146998605950160025344689, −6.18405259507606464087169152397, −5.39297393756147015671675905856, −4.52355432452460168761623664435, −3.19414375359183809376105615785, −1.52782832581302801348174105880, 1.81955210401952935995987153766, 2.55000855242731765819362805204, 3.84653001130115399933245339761, 5.05385001149627792632412189378, 6.23994384736637984784911652788, 7.26892067999449572371863892402, 8.179248265142104393537326013376, 8.933070099298467142499763039899, 10.19429523630352836345527986408, 10.94436107747086797608162783017

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