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Properties

Label 2-510-15.8-c1-0-27
Degree $2$
Conductor $510$
Sign $0.999 - 0.0188i$
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 + (1.18 − 1.26i)3-s + 1.00i·4-s + (2.17 + 0.538i)5-s + (1.73 − 0.0523i)6-s + (0.908 − 0.908i)7-s + (−0.707 + 0.707i)8-s + (−0.181 − 2.99i)9-s + (1.15 + 1.91i)10-s − 2.20i·11-s + (1.26 + 1.18i)12-s + (−1.37 − 1.37i)13-s + 1.28·14-s + (3.25 − 2.09i)15-s − 1.00·16-s + (0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.685 − 0.728i)3-s + 0.500i·4-s + (0.970 + 0.240i)5-s + (0.706 − 0.0213i)6-s + (0.343 − 0.343i)7-s + (−0.250 + 0.250i)8-s + (−0.0603 − 0.998i)9-s + (0.364 + 0.605i)10-s − 0.665i·11-s + (0.364 + 0.342i)12-s + (−0.381 − 0.381i)13-s + 0.343·14-s + (0.840 − 0.541i)15-s − 0.250·16-s + (0.171 + 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(510\)    =    \(2 \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.999 - 0.0188i$
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: \((2,\ 510,\ (\ :1/2),\ 0.999 - 0.0188i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.63073 + 0.0247648i\)
\(L(\frac12)\) \(\approx\) \(2.63073 + 0.0247648i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 + 1.26i)T \)
5 \( 1 + (-2.17 - 0.538i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (-0.908 + 0.908i)T - 7iT^{2} \)
11 \( 1 + 2.20iT - 11T^{2} \)
13 \( 1 + (1.37 + 1.37i)T + 13iT^{2} \)
19 \( 1 - 5.27iT - 19T^{2} \)
23 \( 1 + (4.52 - 4.52i)T - 23iT^{2} \)
29 \( 1 + 9.34T + 29T^{2} \)
31 \( 1 - 4.87T + 31T^{2} \)
37 \( 1 + (-4.06 + 4.06i)T - 37iT^{2} \)
41 \( 1 + 4.56iT - 41T^{2} \)
43 \( 1 + (-6.28 - 6.28i)T + 43iT^{2} \)
47 \( 1 + (-3.07 - 3.07i)T + 47iT^{2} \)
53 \( 1 + (-2.36 + 2.36i)T - 53iT^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 + 1.56T + 61T^{2} \)
67 \( 1 + (3.02 - 3.02i)T - 67iT^{2} \)
71 \( 1 - 7.05iT - 71T^{2} \)
73 \( 1 + (10.1 + 10.1i)T + 73iT^{2} \)
79 \( 1 + 10.6iT - 79T^{2} \)
83 \( 1 + (8.96 - 8.96i)T - 83iT^{2} \)
89 \( 1 - 0.571T + 89T^{2} \)
97 \( 1 + (5.89 - 5.89i)T - 97iT^{2} \)
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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

−10.94616175004620836248146431215, −9.845952559781527277056330147232, −9.022032792330038831734827733923, −7.86250032390813860796148711619, −7.44755389106903651028705908331, −6.07932268477005529916359703052, −5.72337048310780789729905189471, −4.04130096223639014018167456717, −2.93167467497124911963516804173, −1.64396114289984238771844061728, 1.96691934072647851722585911561, 2.73232630637130826964641293884, 4.29425215916523968643795464884, 4.94010414134324348208639126207, 5.95701351416222239213675036023, 7.27059889064062410964768303138, 8.566731692132698909352976428679, 9.360542561794356385687345905328, 9.934744048910936568240143965810, 10.77219154783420404062167179571

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