Properties

Label 2-507-169.4-c1-0-14
Degree $2$
Conductor $507$
Sign $0.289 + 0.957i$
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  + (−2.60 − 0.104i)2-s + (−0.278 + 0.960i)3-s + (4.76 + 0.384i)4-s + (−0.705 − 0.267i)5-s + (0.824 − 2.47i)6-s + (0.861 + 2.02i)7-s + (−7.19 − 0.873i)8-s + (−0.845 − 0.534i)9-s + (1.80 + 0.769i)10-s + (−2.17 − 3.44i)11-s + (−1.69 + 4.47i)12-s + (0.939 − 3.48i)13-s + (−2.02 − 5.35i)14-s + (0.453 − 0.602i)15-s + (9.17 + 1.49i)16-s + (3.69 − 1.57i)17-s + ⋯
L(s)  = 1  + (−1.83 − 0.0741i)2-s + (−0.160 + 0.554i)3-s + (2.38 + 0.192i)4-s + (−0.315 − 0.119i)5-s + (0.336 − 1.00i)6-s + (0.325 + 0.764i)7-s + (−2.54 − 0.308i)8-s + (−0.281 − 0.178i)9-s + (0.571 + 0.243i)10-s + (−0.656 − 1.03i)11-s + (−0.489 + 1.29i)12-s + (0.260 − 0.965i)13-s + (−0.542 − 1.43i)14-s + (0.116 − 0.155i)15-s + (2.29 + 0.372i)16-s + (0.897 − 0.382i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.289 + 0.957i$
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: \((2,\ 507,\ (\ :1/2),\ 0.289 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.286522 - 0.212701i\)
\(L(\frac12)\) \(\approx\) \(0.286522 - 0.212701i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.939 + 3.48i)T \)
good2 \( 1 + (2.60 + 0.104i)T + (1.99 + 0.160i)T^{2} \)
5 \( 1 + (0.705 + 0.267i)T + (3.74 + 3.31i)T^{2} \)
7 \( 1 + (-0.861 - 2.02i)T + (-4.84 + 5.04i)T^{2} \)
11 \( 1 + (2.17 + 3.44i)T + (-4.71 + 9.93i)T^{2} \)
17 \( 1 + (-3.69 + 1.57i)T + (11.7 - 12.2i)T^{2} \)
19 \( 1 + (6.05 - 3.49i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.272 + 0.472i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.236 - 5.87i)T + (-28.9 - 2.33i)T^{2} \)
31 \( 1 + (4.18 + 4.72i)T + (-3.73 + 30.7i)T^{2} \)
37 \( 1 + (-9.47 + 1.93i)T + (34.0 - 14.5i)T^{2} \)
41 \( 1 + (1.15 + 0.335i)T + (34.6 + 21.9i)T^{2} \)
43 \( 1 + (-2.26 + 11.1i)T + (-39.5 - 16.8i)T^{2} \)
47 \( 1 + (1.43 - 0.987i)T + (16.6 - 43.9i)T^{2} \)
53 \( 1 + (-0.291 + 2.39i)T + (-51.4 - 12.6i)T^{2} \)
59 \( 1 + (2.22 + 13.7i)T + (-55.9 + 18.6i)T^{2} \)
61 \( 1 + (-6.88 + 5.17i)T + (16.9 - 58.5i)T^{2} \)
67 \( 1 + (-0.0808 - 1.00i)T + (-66.1 + 10.7i)T^{2} \)
71 \( 1 + (1.91 - 1.84i)T + (2.85 - 70.9i)T^{2} \)
73 \( 1 + (-0.0218 + 0.0416i)T + (-41.4 - 60.0i)T^{2} \)
79 \( 1 + (0.892 + 1.29i)T + (-28.0 + 73.8i)T^{2} \)
83 \( 1 + (-3.76 + 15.2i)T + (-73.4 - 38.5i)T^{2} \)
89 \( 1 + (-3.51 - 2.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.831 - 0.679i)T + (19.4 + 95.0i)T^{2} \)
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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

−10.59948950281341724026918682262, −9.855405867590155403832596215408, −8.836212912530695374366357660400, −8.260312097722617126751486576596, −7.67688647775403049386864769266, −6.17559617228061062390173026714, −5.44184857368248352986445424905, −3.50711413557166654870028196048, −2.26300925391248700239638293886, −0.41484679488921540750620860702, 1.30844576907980996701709808853, 2.43516643908312921141918954471, 4.33782357665206577634866102588, 6.07723978202885098549517988773, 7.03491293161663536455752890298, 7.59050229918738015007319264430, 8.265404120400687680104907341618, 9.323125252101898991002585212278, 10.12067323932637800634222437282, 10.95050122033247706746177468424

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