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Properties

Label 2-510-15.8-c1-0-4
Degree $2$
Conductor $510$
Sign $-0.998 - 0.0618i$
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 + (0.292 + 1.70i)3-s + 1.00i·4-s + (−0.707 − 2.12i)5-s + (−0.999 + 1.41i)6-s + (−3 + 3i)7-s + (−0.707 + 0.707i)8-s + (−2.82 + i)9-s + (0.999 − 2i)10-s + 1.41i·11-s + (−1.70 + 0.292i)12-s + (−2 − 2i)13-s − 4.24·14-s + (3.41 − 1.82i)15-s − 1.00·16-s + (−0.707 − 0.707i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.169 + 0.985i)3-s + 0.500i·4-s + (−0.316 − 0.948i)5-s + (−0.408 + 0.577i)6-s + (−1.13 + 1.13i)7-s + (−0.250 + 0.250i)8-s + (−0.942 + 0.333i)9-s + (0.316 − 0.632i)10-s + 0.426i·11-s + (−0.492 + 0.0845i)12-s + (−0.554 − 0.554i)13-s − 1.13·14-s + (0.881 − 0.472i)15-s − 0.250·16-s + (−0.171 − 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(510\)    =    \(2 \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.998 - 0.0618i$
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: \((2,\ 510,\ (\ :1/2),\ -0.998 - 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0349625 + 1.12918i\)
\(L(\frac12)\) \(\approx\) \(0.0349625 + 1.12918i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 - 1.70i)T \)
5 \( 1 + (0.707 + 2.12i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + (2 + 2i)T + 13iT^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + (2 - 2i)T - 37iT^{2} \)
41 \( 1 - 8.48iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (2.82 + 2.82i)T + 47iT^{2} \)
53 \( 1 + (-5.65 + 5.65i)T - 53iT^{2} \)
59 \( 1 + 9.89T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + (-10 + 10i)T - 67iT^{2} \)
71 \( 1 - 11.3iT - 71T^{2} \)
73 \( 1 + (3 + 3i)T + 73iT^{2} \)
79 \( 1 - 14iT - 79T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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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

−11.65996609062362033171880452393, −9.992592392349487341500001118567, −9.700553157072122388932490185858, −8.519705286139752625561745608488, −8.075408458950135062980248563009, −6.46021880493944077043180218039, −5.56272523285309459113258261011, −4.81953374784323550765130951960, −3.73413770322209102508212922207, −2.67859133565610921714790538138, 0.53424918103776138478706989268, 2.52926617754969113857136637879, 3.27113013812821530091337727714, 4.44738002019127870663619622683, 6.13915091699981493125103714929, 6.85261370781406499158019558396, 7.30421339375458048749440212658, 8.696826708120855493386382771360, 9.813403364338584597918642420865, 10.65097001300400552525001082117

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