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Properties

Label 2-510-15.8-c1-0-15
Degree 22
Conductor 510510
Sign 0.522+0.852i0.522 + 0.852i
Analytic cond. 4.072374.07237
Root an. cond. 2.018012.01801
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(510s/2 ΓC(s) L(s)=((0.522+0.852i) Λ‾(2−s)\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}
Λ(s)=(510s/2 ΓC(s+1/2) L(s)=((0.522+0.852i) Λ‾(1−s)\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: 22
Conductor: 510510    =    2⋅3⋅5⋅172 \cdot 3 \cdot 5 \cdot 17
Sign: 0.522+0.852i0.522 + 0.852i
Analytic conductor: 4.072374.07237
Root analytic conductor: 2.018012.01801
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ510(443,⋅)\chi_{510} (443, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 510, ( :1/2), 0.522+0.852i)(2,\ 510,\ (\ :1/2),\ 0.522 + 0.852i)

Particular Values

L(1)L(1) ≈\approx 0.951576−0.532780i0.951576 - 0.532780i
L(12)L(\frac12) ≈\approx 0.951576−0.532780i0.951576 - 0.532780i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
3 1+(1.41+i)T 1 + (1.41 + i)T
5 1+(−2.21+0.330i)T 1 + (-2.21 + 0.330i)T
17 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good7 1+(−0.532+0.532i)T−7iT2 1 + (-0.532 + 0.532i)T - 7iT^{2}
11 1+1.41iT−11T2 1 + 1.41iT - 11T^{2}
13 1+(−3.59−3.59i)T+13iT2 1 + (-3.59 - 3.59i)T + 13iT^{2}
19 1−7.17iT−19T2 1 - 7.17iT - 19T^{2}
23 1+(−6.47+6.47i)T−23iT2 1 + (-6.47 + 6.47i)T - 23iT^{2}
29 1+1.23T+29T2 1 + 1.23T + 29T^{2}
31 1+4.51T+31T2 1 + 4.51T + 31T^{2}
37 1+(−6.30+6.30i)T−37iT2 1 + (-6.30 + 6.30i)T - 37iT^{2}
41 1+4.69iT−41T2 1 + 4.69iT - 41T^{2}
43 1+(4.83+4.83i)T+43iT2 1 + (4.83 + 4.83i)T + 43iT^{2}
47 1+(6+6i)T+47iT2 1 + (6 + 6i)T + 47iT^{2}
53 1+(0.660−0.660i)T−53iT2 1 + (0.660 - 0.660i)T - 53iT^{2}
59 1−14.8T+59T2 1 - 14.8T + 59T^{2}
61 1+2.79T+61T2 1 + 2.79T + 61T^{2}
67 1+(−6.49+6.49i)T−67iT2 1 + (-6.49 + 6.49i)T - 67iT^{2}
71 1−8.43iT−71T2 1 - 8.43iT - 71T^{2}
73 1+(−1.32−1.32i)T+73iT2 1 + (-1.32 - 1.32i)T + 73iT^{2}
79 1+4.77iT−79T2 1 + 4.77iT - 79T^{2}
83 1+(8.83−8.83i)T−83iT2 1 + (8.83 - 8.83i)T - 83iT^{2}
89 1−2.73T+89T2 1 - 2.73T + 89T^{2}
97 1+(10.7−10.7i)T−97iT2 1 + (10.7 - 10.7i)T - 97iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line