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Properties

Label 2-507-39.2-c1-0-0
Degree $2$
Conductor $507$
Sign $-0.911 + 0.411i$
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  + (−0.389 + 1.45i)2-s + (0.239 − 1.71i)3-s + (−0.232 − 0.133i)4-s + (−1.06 − 1.06i)5-s + (2.40 + 1.01i)6-s + (−1.36 + 0.366i)7-s + (−1.84 + 1.84i)8-s + (−2.88 − 0.820i)9-s + (1.96 − 1.13i)10-s + (−3.97 − 1.06i)11-s + (−0.285 + 0.366i)12-s − 2.12i·14-s + (−2.08 + 1.57i)15-s + (−2.23 − 3.86i)16-s + (−2.51 + 4.36i)17-s + (2.31 − 3.87i)18-s + ⋯
L(s)  = 1  + (−0.275 + 1.02i)2-s + (0.138 − 0.990i)3-s + (−0.116 − 0.0669i)4-s + (−0.476 − 0.476i)5-s + (0.980 + 0.415i)6-s + (−0.516 + 0.138i)7-s + (−0.652 + 0.652i)8-s + (−0.961 − 0.273i)9-s + (0.621 − 0.358i)10-s + (−1.19 − 0.321i)11-s + (−0.0823 + 0.105i)12-s − 0.569i·14-s + (−0.537 + 0.405i)15-s + (−0.558 − 0.966i)16-s + (−0.611 + 1.05i)17-s + (0.546 − 0.913i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.911 + 0.411i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.911 + 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00850127 - 0.0395298i\)
\(L(\frac12)\) \(\approx\) \(0.00850127 - 0.0395298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.239 + 1.71i)T \)
13 \( 1 \)
good2 \( 1 + (0.389 - 1.45i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (1.06 + 1.06i)T + 5iT^{2} \)
7 \( 1 + (1.36 - 0.366i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3.97 + 1.06i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.51 - 4.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 3.73i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.20 - 3.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.46 - 2.46i)T - 31iT^{2} \)
37 \( 1 + (-1.40 + 5.23i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.45 + 5.42i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.90 + 1.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.25 + 4.25i)T - 47iT^{2} \)
53 \( 1 + 0.779iT - 53T^{2} \)
59 \( 1 + (0.779 + 2.90i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.73 - 1.53i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.90 - 0.779i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.901 + 0.901i)T + 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (2.90 + 2.90i)T + 83iT^{2} \)
89 \( 1 + (9.01 + 2.41i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.437 + 1.63i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.52237720727059488849598553299, −10.59061088494360466166059536953, −9.118793029119718815274578382147, −8.382809256037681774750216656168, −7.81697309759491730743092461917, −7.01265502568324852076093482896, −6.04304461061674270011083326412, −5.36268736031199872697679734951, −3.53388519986767461516501726760, −2.19470052769017580201637403066, 0.02317969489445553938115078186, 2.52541453533409067604436200435, 3.15908906436401552252714489642, 4.30826097579240401533451548668, 5.48948636716612827420938244459, 6.81211804365561228205414003598, 7.81734874854293331452087211522, 9.161092719978273950401469426355, 9.685147129551993754038609811690, 10.45712312408807282681026449669

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