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Properties

Label 2-510-17.8-c1-0-3
Degree $2$
Conductor $510$
Sign $0.915 + 0.402i$
Analytic cond. $4.07237$
Root an. cond. $2.01801$
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  + (−0.707 + 0.707i)2-s + (−0.382 + 0.923i)3-s − 1.00i·4-s + (−0.923 − 0.382i)5-s + (−0.382 − 0.923i)6-s + (−0.566 + 0.234i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + (−0.873 − 2.10i)11-s + (0.923 + 0.382i)12-s − 4.76i·13-s + (0.234 − 0.566i)14-s + (0.707 − 0.707i)15-s − 1.00·16-s + (3.63 + 1.94i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.220 + 0.533i)3-s − 0.500i·4-s + (−0.413 − 0.171i)5-s + (−0.156 − 0.377i)6-s + (−0.214 + 0.0886i)7-s + (0.250 + 0.250i)8-s + (−0.235 − 0.235i)9-s + (0.292 − 0.121i)10-s + (−0.263 − 0.635i)11-s + (0.266 + 0.110i)12-s − 1.32i·13-s + (0.0627 − 0.151i)14-s + (0.182 − 0.182i)15-s − 0.250·16-s + (0.882 + 0.470i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(510\)    =    \(2 \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.915 + 0.402i$
Analytic conductor: \(4.07237\)
Root analytic conductor: \(2.01801\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{510} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 510,\ (\ :1/2),\ 0.915 + 0.402i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.766967 - 0.161228i\)
\(L(\frac12)\) \(\approx\) \(0.766967 - 0.161228i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 + (-3.63 - 1.94i)T \)
good7 \( 1 + (0.566 - 0.234i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.873 + 2.10i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 4.76iT - 13T^{2} \)
19 \( 1 + (-2.82 + 2.82i)T - 19iT^{2} \)
23 \( 1 + (-1.31 - 3.16i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (8.78 + 3.63i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-3.61 + 8.73i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-4.52 + 10.9i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-6.83 + 2.83i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-7.81 - 7.81i)T + 43iT^{2} \)
47 \( 1 - 8.45iT - 47T^{2} \)
53 \( 1 + (-1.86 + 1.86i)T - 53iT^{2} \)
59 \( 1 + (6.16 + 6.16i)T + 59iT^{2} \)
61 \( 1 + (-8.52 + 3.53i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 2.74T + 67T^{2} \)
71 \( 1 + (1.48 - 3.57i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (9.33 + 3.86i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (3.54 + 8.56i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-4.65 + 4.65i)T - 83iT^{2} \)
89 \( 1 + 2.68iT - 89T^{2} \)
97 \( 1 + (12.9 + 5.35i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.92808558035890320312437589698, −9.705076105112366652693141499593, −9.256940861433745935192547215800, −7.87891502301790787961471020058, −7.65029856439171059512988517494, −5.90693859324470600522967537774, −5.57991519051908261203241164670, −4.15328330211796491323284076570, −2.94511727516509401449800233909, −0.62681998414834930951973800788, 1.37572962517809413070561298872, 2.78492565695092381970590364931, 4.04154931552769447284919665804, 5.29880263345125083540084855722, 6.74921018522804896488591228890, 7.31189172655848412231048494771, 8.264579491770052859089505229918, 9.289846708401385407368344957706, 10.09442287503110482091402657546, 11.01299799926325008124991828098

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