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Properties

Label 2-605-55.32-c1-0-8
Degree 22
Conductor 605605
Sign −0.655−0.755i-0.655 - 0.755i
Analytic cond. 4.830944.83094
Root an. cond. 2.197942.19794
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  + (0.345 − 0.345i)2-s + (−0.805 + 0.805i)3-s + 1.76i·4-s + (−2.18 − 0.490i)5-s + 0.556i·6-s + (2.06 − 2.06i)7-s + (1.29 + 1.29i)8-s + 1.70i·9-s + (−0.922 + 0.583i)10-s + (−1.41 − 1.41i)12-s + (2.06 + 2.06i)13-s − 1.42i·14-s + (2.15 − 1.36i)15-s − 2.62·16-s + (−3.72 + 3.72i)17-s + (0.587 + 0.587i)18-s + ⋯
L(s)  = 1  + (0.244 − 0.244i)2-s + (−0.465 + 0.465i)3-s + 0.880i·4-s + (−0.975 − 0.219i)5-s + 0.227i·6-s + (0.779 − 0.779i)7-s + (0.459 + 0.459i)8-s + 0.567i·9-s + (−0.291 + 0.184i)10-s + (−0.409 − 0.409i)12-s + (0.573 + 0.573i)13-s − 0.380i·14-s + (0.555 − 0.351i)15-s − 0.656·16-s + (−0.903 + 0.903i)17-s + (0.138 + 0.138i)18-s + ⋯

Functional equation

Λ(s)=(605s/2 ΓC(s) L(s)=((−0.655−0.755i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(605s/2 ΓC(s+1/2) L(s)=((−0.655−0.755i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 605605    =    5⋅1125 \cdot 11^{2}
Sign: −0.655−0.755i-0.655 - 0.755i
Analytic conductor: 4.830944.83094
Root analytic conductor: 2.197942.19794
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ605(362,⋅)\chi_{605} (362, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 605, ( :1/2), −0.655−0.755i)(2,\ 605,\ (\ :1/2),\ -0.655 - 0.755i)

Particular Values

L(1)L(1) ≈\approx 0.364032+0.797404i0.364032 + 0.797404i
L(12)L(\frac12) ≈\approx 0.364032+0.797404i0.364032 + 0.797404i
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)
bad5 1+(2.18+0.490i)T 1 + (2.18 + 0.490i)T
11 1 1
good2 1+(−0.345+0.345i)T−2iT2 1 + (-0.345 + 0.345i)T - 2iT^{2}
3 1+(0.805−0.805i)T−3iT2 1 + (0.805 - 0.805i)T - 3iT^{2}
7 1+(−2.06+2.06i)T−7iT2 1 + (-2.06 + 2.06i)T - 7iT^{2}
13 1+(−2.06−2.06i)T+13iT2 1 + (-2.06 - 2.06i)T + 13iT^{2}
17 1+(3.72−3.72i)T−17iT2 1 + (3.72 - 3.72i)T - 17iT^{2}
19 1+4.09T+19T2 1 + 4.09T + 19T^{2}
23 1+(2.12−2.12i)T−23iT2 1 + (2.12 - 2.12i)T - 23iT^{2}
29 1+2.64T+29T2 1 + 2.64T + 29T^{2}
31 1+6.74T+31T2 1 + 6.74T + 31T^{2}
37 1+(−0.822−0.822i)T+37iT2 1 + (-0.822 - 0.822i)T + 37iT^{2}
41 1−3.55iT−41T2 1 - 3.55iT - 41T^{2}
43 1+(−5.07−5.07i)T+43iT2 1 + (-5.07 - 5.07i)T + 43iT^{2}
47 1+(2.60+2.60i)T+47iT2 1 + (2.60 + 2.60i)T + 47iT^{2}
53 1+(−1.04+1.04i)T−53iT2 1 + (-1.04 + 1.04i)T - 53iT^{2}
59 1−1.60iT−59T2 1 - 1.60iT - 59T^{2}
61 1+6.93iT−61T2 1 + 6.93iT - 61T^{2}
67 1+(−1.31−1.31i)T+67iT2 1 + (-1.31 - 1.31i)T + 67iT^{2}
71 1−2.87T+71T2 1 - 2.87T + 71T^{2}
73 1+(−10.4−10.4i)T+73iT2 1 + (-10.4 - 10.4i)T + 73iT^{2}
79 1−15.2T+79T2 1 - 15.2T + 79T^{2}
83 1+(−4.72−4.72i)T+83iT2 1 + (-4.72 - 4.72i)T + 83iT^{2}
89 1+11.1iT−89T2 1 + 11.1iT - 89T^{2}
97 1+(−7.11−7.11i)T+97iT2 1 + (-7.11 - 7.11i)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

−11.16247786154390228439797232509, −10.59649942366885964335123782065, −9.064511470387725263174932623561, −8.135044764775652018507409941513, −7.70361167023791057737059564687, −6.57407986342850975244092189770, −5.04031093887339567003333847054, −4.19123476431040735639791759086, −3.82108409522325554337362881181, −1.96113431573570686827673205155, 0.47036276895506745524001796732, 2.10684142602976516850681637972, 3.83705277468178342951083421130, 4.91283063382061315906858942745, 5.82037488788201848144557793730, 6.63196658234919471653227618797, 7.47213751918136902259537620822, 8.579245430091363723070319941616, 9.316211775566107937910359139178, 10.79590774757326125355467316973

Graph of the ZZ-function along the critical line