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Properties

Label 2-510-15.2-c1-0-4
Degree $2$
Conductor $510$
Sign $-0.995 + 0.0921i$
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 + (0.209 + 1.71i)3-s − 1.00i·4-s + (−1.22 + 1.86i)5-s + (−1.36 − 1.06i)6-s + (1.96 + 1.96i)7-s + (0.707 + 0.707i)8-s + (−2.91 + 0.719i)9-s + (−0.450 − 2.19i)10-s + 2.52i·11-s + (1.71 − 0.209i)12-s + (0.989 − 0.989i)13-s − 2.78·14-s + (−3.46 − 1.72i)15-s − 1.00·16-s + (−0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.120 + 0.992i)3-s − 0.500i·4-s + (−0.550 + 0.835i)5-s + (−0.556 − 0.435i)6-s + (0.743 + 0.743i)7-s + (0.250 + 0.250i)8-s + (−0.970 + 0.239i)9-s + (−0.142 − 0.692i)10-s + 0.762i·11-s + (0.496 − 0.0604i)12-s + (0.274 − 0.274i)13-s − 0.743·14-s + (−0.895 − 0.445i)15-s − 0.250·16-s + (−0.171 + 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(510\)    =    \(2 \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.995 + 0.0921i$
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: \((2,\ 510,\ (\ :1/2),\ -0.995 + 0.0921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0418637 - 0.906452i\)
\(L(\frac12)\) \(\approx\) \(0.0418637 - 0.906452i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.209 - 1.71i)T \)
5 \( 1 + (1.22 - 1.86i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-1.96 - 1.96i)T + 7iT^{2} \)
11 \( 1 - 2.52iT - 11T^{2} \)
13 \( 1 + (-0.989 + 0.989i)T - 13iT^{2} \)
19 \( 1 + 1.20iT - 19T^{2} \)
23 \( 1 + (-2.53 - 2.53i)T + 23iT^{2} \)
29 \( 1 + 4.48T + 29T^{2} \)
31 \( 1 + 2.67T + 31T^{2} \)
37 \( 1 + (0.953 + 0.953i)T + 37iT^{2} \)
41 \( 1 - 8.82iT - 41T^{2} \)
43 \( 1 + (-0.0549 + 0.0549i)T - 43iT^{2} \)
47 \( 1 + (-8.35 + 8.35i)T - 47iT^{2} \)
53 \( 1 + (7.21 + 7.21i)T + 53iT^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 - 7.63T + 61T^{2} \)
67 \( 1 + (-9.61 - 9.61i)T + 67iT^{2} \)
71 \( 1 - 6.19iT - 71T^{2} \)
73 \( 1 + (4.99 - 4.99i)T - 73iT^{2} \)
79 \( 1 - 6.89iT - 79T^{2} \)
83 \( 1 + (6.01 + 6.01i)T + 83iT^{2} \)
89 \( 1 - 8.16T + 89T^{2} \)
97 \( 1 + (-12.1 - 12.1i)T + 97iT^{2} \)
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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

−11.22882724324155287997850664829, −10.39682177443685588841815068232, −9.552807701698655339529070959326, −8.669768419419634105100380099493, −7.909690701570347534265105181537, −6.95202309458128493998733916451, −5.72536749412568502021630939223, −4.84918877032421119053694560076, −3.66493983111407222845689905792, −2.29394185561494649131840896362, 0.64165197994870659502679312948, 1.74757674314925537426281531446, 3.37671923171299736088767812356, 4.53007210905309583767083919886, 5.84830762763058324632341711858, 7.21018644676950990686719768014, 7.81644219049755096772464763005, 8.629479956588222731583698476653, 9.241430509178308902878595586525, 10.87006690375124054180147538963

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