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Properties

Label 2-507-169.4-c1-0-2
Degree $2$
Conductor $507$
Sign $-0.996 - 0.0810i$
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 − 0.0558i)2-s + (−0.278 + 0.960i)3-s + (−0.0758 − 0.00612i)4-s + (−1.08 − 0.410i)5-s + (0.439 − 1.31i)6-s + (−0.389 − 0.913i)7-s + (2.85 + 0.347i)8-s + (−0.845 − 0.534i)9-s + (1.47 + 0.630i)10-s + (1.94 + 3.07i)11-s + (0.0269 − 0.0711i)12-s + (3.40 − 1.18i)13-s + (0.488 + 1.28i)14-s + (0.696 − 0.926i)15-s + (−3.79 − 0.616i)16-s + (−4.47 + 1.90i)17-s + ⋯
L(s)  = 1  + (−0.979 − 0.0394i)2-s + (−0.160 + 0.554i)3-s + (−0.0379 − 0.00306i)4-s + (−0.484 − 0.183i)5-s + (0.179 − 0.537i)6-s + (−0.147 − 0.345i)7-s + (1.01 + 0.122i)8-s + (−0.281 − 0.178i)9-s + (0.467 + 0.199i)10-s + (0.587 + 0.928i)11-s + (0.00779 − 0.0205i)12-s + (0.944 − 0.328i)13-s + (0.130 + 0.344i)14-s + (0.179 − 0.239i)15-s + (−0.948 − 0.154i)16-s + (−1.08 + 0.462i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.996 - 0.0810i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.996 - 0.0810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00576731 + 0.142127i\)
\(L(\frac12)\) \(\approx\) \(0.00576731 + 0.142127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.278 - 0.960i)T \)
13 \( 1 + (-3.40 + 1.18i)T \)
good2 \( 1 + (1.38 + 0.0558i)T + (1.99 + 0.160i)T^{2} \)
5 \( 1 + (1.08 + 0.410i)T + (3.74 + 3.31i)T^{2} \)
7 \( 1 + (0.389 + 0.913i)T + (-4.84 + 5.04i)T^{2} \)
11 \( 1 + (-1.94 - 3.07i)T + (-4.71 + 9.93i)T^{2} \)
17 \( 1 + (4.47 - 1.90i)T + (11.7 - 12.2i)T^{2} \)
19 \( 1 + (3.01 - 1.74i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.61 - 4.53i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0493 + 1.22i)T + (-28.9 - 2.33i)T^{2} \)
31 \( 1 + (4.88 + 5.51i)T + (-3.73 + 30.7i)T^{2} \)
37 \( 1 + (2.26 - 0.461i)T + (34.0 - 14.5i)T^{2} \)
41 \( 1 + (3.65 + 1.05i)T + (34.6 + 21.9i)T^{2} \)
43 \( 1 + (-0.850 + 4.16i)T + (-39.5 - 16.8i)T^{2} \)
47 \( 1 + (10.5 - 7.29i)T + (16.6 - 43.9i)T^{2} \)
53 \( 1 + (-0.163 + 1.35i)T + (-51.4 - 12.6i)T^{2} \)
59 \( 1 + (-1.11 - 6.88i)T + (-55.9 + 18.6i)T^{2} \)
61 \( 1 + (12.0 - 9.08i)T + (16.9 - 58.5i)T^{2} \)
67 \( 1 + (0.176 + 2.18i)T + (-66.1 + 10.7i)T^{2} \)
71 \( 1 + (8.03 - 7.72i)T + (2.85 - 70.9i)T^{2} \)
73 \( 1 + (3.63 - 6.93i)T + (-41.4 - 60.0i)T^{2} \)
79 \( 1 + (-4.09 - 5.93i)T + (-28.0 + 73.8i)T^{2} \)
83 \( 1 + (0.464 - 1.88i)T + (-73.4 - 38.5i)T^{2} \)
89 \( 1 + (-0.800 - 0.462i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.97 + 4.87i)T + (19.4 + 95.0i)T^{2} \)
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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

−11.04752079560163247411318929638, −10.30646135503976712641334795269, −9.580616167679561341864158093748, −8.751091633031711699411344906338, −8.035642136974568633150354398414, −7.03008606490572437470437225463, −5.83881443852078635648981894020, −4.35191262503310904190346622916, −3.92341593478484866621730359199, −1.72294616972153775490581678914, 0.12316914119419823161472961435, 1.74577781890473184283901346767, 3.48947111621889151473478405149, 4.70952735518449827993690051525, 6.21523481707423768904594068726, 6.89094513020816034256375802538, 8.016802646606980613979044817809, 8.749341018734346646864268865178, 9.208053322606370911767695094444, 10.63843089096298165782630563696

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