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Properties

Label 2-510-15.8-c1-0-15
Degree $2$
Conductor $510$
Sign $0.522 + 0.852i$
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 + (−1.41 − i)3-s + 1.00i·4-s + (2.21 − 0.330i)5-s + (0.292 + 1.70i)6-s + (0.532 − 0.532i)7-s + (0.707 − 0.707i)8-s + (1.00 + 2.82i)9-s + (−1.79 − 1.33i)10-s − 1.41i·11-s + (1.00 − 1.41i)12-s + (3.59 + 3.59i)13-s − 0.753·14-s + (−3.45 − 1.74i)15-s − 1.00·16-s + (0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.816 − 0.577i)3-s + 0.500i·4-s + (0.989 − 0.147i)5-s + (0.119 + 0.696i)6-s + (0.201 − 0.201i)7-s + (0.250 − 0.250i)8-s + (0.333 + 0.942i)9-s + (−0.568 − 0.420i)10-s − 0.426i·11-s + (0.288 − 0.408i)12-s + (0.996 + 0.996i)13-s − 0.201·14-s + (−0.892 − 0.450i)15-s − 0.250·16-s + (0.171 + 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(510\)    =    \(2 \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(4.07237\)
Root analytic conductor: \(2.01801\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{510} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 510,\ (\ :1/2),\ 0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951576 - 0.532780i\)
\(L(\frac12)\) \(\approx\) \(0.951576 - 0.532780i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + i)T \)
5 \( 1 + (-2.21 + 0.330i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (-0.532 + 0.532i)T - 7iT^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + (-3.59 - 3.59i)T + 13iT^{2} \)
19 \( 1 - 7.17iT - 19T^{2} \)
23 \( 1 + (-6.47 + 6.47i)T - 23iT^{2} \)
29 \( 1 + 1.23T + 29T^{2} \)
31 \( 1 + 4.51T + 31T^{2} \)
37 \( 1 + (-6.30 + 6.30i)T - 37iT^{2} \)
41 \( 1 + 4.69iT - 41T^{2} \)
43 \( 1 + (4.83 + 4.83i)T + 43iT^{2} \)
47 \( 1 + (6 + 6i)T + 47iT^{2} \)
53 \( 1 + (0.660 - 0.660i)T - 53iT^{2} \)
59 \( 1 - 14.8T + 59T^{2} \)
61 \( 1 + 2.79T + 61T^{2} \)
67 \( 1 + (-6.49 + 6.49i)T - 67iT^{2} \)
71 \( 1 - 8.43iT - 71T^{2} \)
73 \( 1 + (-1.32 - 1.32i)T + 73iT^{2} \)
79 \( 1 + 4.77iT - 79T^{2} \)
83 \( 1 + (8.83 - 8.83i)T - 83iT^{2} \)
89 \( 1 - 2.73T + 89T^{2} \)
97 \( 1 + (10.7 - 10.7i)T - 97iT^{2} \)
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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

−10.81044765710181406142985538351, −10.09682533367883470464463969551, −9.000141482959382679185344062740, −8.230004685249573338997156825535, −6.99656774198914734443247858321, −6.20063898470300525523863461268, −5.28776715153488829075120662367, −3.91083591309264782997747382449, −2.14443296224122856093107772874, −1.14364697608481355145141205339, 1.20301184455750855244791423486, 3.10450695873889055521030188420, 4.87061478386813556376300736298, 5.47040331301032000674636515076, 6.38989991044200515595775407590, 7.21027611562741004367830158289, 8.567491626107084120520519121114, 9.467925007889555538757796716526, 9.958385326308910840039387375432, 11.09898139338879660392914174607

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