↑

Properties

Label 2-510-15.2-c1-0-23
Degree $2$
Conductor $510$
Sign $-0.837 + 0.547i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−1.70 − 0.318i)3-s − 1.00i·4-s + (2.18 − 0.466i)5-s + (1.42 − 0.978i)6-s + (−2.68 − 2.68i)7-s + (0.707 + 0.707i)8-s + (2.79 + 1.08i)9-s + (−1.21 + 1.87i)10-s − 4.10i·11-s + (−0.318 + 1.70i)12-s + (−4.65 + 4.65i)13-s + 3.79·14-s + (−3.87 + 0.0979i)15-s − 1.00·16-s + (−0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.982 − 0.183i)3-s − 0.500i·4-s + (0.978 − 0.208i)5-s + (0.583 − 0.399i)6-s + (−1.01 − 1.01i)7-s + (0.250 + 0.250i)8-s + (0.932 + 0.361i)9-s + (−0.384 + 0.593i)10-s − 1.23i·11-s + (−0.0918 + 0.491i)12-s + (−1.29 + 1.29i)13-s + 1.01·14-s + (−0.999 + 0.0252i)15-s − 0.250·16-s + (−0.171 + 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(510\)    =    \(2 \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.837 + 0.547i$
Analytic conductor: \(4.07237\)
Root analytic conductor: \(2.01801\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{510} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 510,\ (\ :1/2),\ -0.837 + 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0713204 - 0.239499i\)
\(L(\frac12)\) \(\approx\) \(0.0713204 - 0.239499i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.318i)T \)
5 \( 1 + (-2.18 + 0.466i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (2.68 + 2.68i)T + 7iT^{2} \)
11 \( 1 + 4.10iT - 11T^{2} \)
13 \( 1 + (4.65 - 4.65i)T - 13iT^{2} \)
19 \( 1 - 2.50iT - 19T^{2} \)
23 \( 1 + (-0.106 - 0.106i)T + 23iT^{2} \)
29 \( 1 + 5.15T + 29T^{2} \)
31 \( 1 + 9.65T + 31T^{2} \)
37 \( 1 + (6.88 + 6.88i)T + 37iT^{2} \)
41 \( 1 + 2.12iT - 41T^{2} \)
43 \( 1 + (-4.80 + 4.80i)T - 43iT^{2} \)
47 \( 1 + (3.75 - 3.75i)T - 47iT^{2} \)
53 \( 1 + (3.13 + 3.13i)T + 53iT^{2} \)
59 \( 1 + 3.38T + 59T^{2} \)
61 \( 1 + 4.42T + 61T^{2} \)
67 \( 1 + (-7.73 - 7.73i)T + 67iT^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + (2.67 - 2.67i)T - 73iT^{2} \)
79 \( 1 - 8.13iT - 79T^{2} \)
83 \( 1 + (3.24 + 3.24i)T + 83iT^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 + (5.96 + 5.96i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.48747900145155126686274839667, −9.626144133082701102370330193181, −9.053015315396162136604110949520, −7.45563166175038584750830205182, −6.83659242754900753779141647202, −6.02677316594693611623335773147, −5.24205381329920749600615045295, −3.88587652725806331904305952748, −1.82599810791216654575549725501, −0.18499277752561908906665717308, 2.00431407663940957977197321645, 3.09741450118128093695955246937, 4.89413979136942808224440197282, 5.61841109328505878578239438052, 6.69220079698983931880830165177, 7.46707736939675331803765397095, 9.166497347819918593744435214472, 9.661318179617334509267451538799, 10.20420745013512798610976726395, 11.09298288406868398057599797746

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