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Properties

Label 2-425-85.64-c1-0-13
Degree 22
Conductor 425425
Sign 0.587+0.809i0.587 + 0.809i
Analytic cond. 3.393643.39364
Root an. cond. 1.842181.84218
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  − 1.12·2-s + (1.75 − 1.75i)3-s − 0.729·4-s + (−1.97 + 1.97i)6-s + (1.72 + 1.72i)7-s + 3.07·8-s − 3.16i·9-s + (2.57 − 2.57i)11-s + (−1.28 + 1.28i)12-s + 3.64i·13-s + (−1.94 − 1.94i)14-s − 2.00·16-s + (2.79 − 3.03i)17-s + 3.56i·18-s − 2.61i·19-s + ⋯
L(s)  = 1  − 0.796·2-s + (1.01 − 1.01i)3-s − 0.364·4-s + (−0.807 + 0.807i)6-s + (0.652 + 0.652i)7-s + 1.08·8-s − 1.05i·9-s + (0.775 − 0.775i)11-s + (−0.369 + 0.369i)12-s + 1.01i·13-s + (−0.520 − 0.520i)14-s − 0.502·16-s + (0.677 − 0.735i)17-s + 0.840i·18-s − 0.599i·19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((0.587+0.809i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((0.587+0.809i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: 0.587+0.809i0.587 + 0.809i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(149,⋅)\chi_{425} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), 0.587+0.809i)(2,\ 425,\ (\ :1/2),\ 0.587 + 0.809i)

Particular Values

L(1)L(1) ≈\approx 1.13632−0.579471i1.13632 - 0.579471i
L(12)L(\frac12) ≈\approx 1.13632−0.579471i1.13632 - 0.579471i
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 1
17 1+(−2.79+3.03i)T 1 + (-2.79 + 3.03i)T
good2 1+1.12T+2T2 1 + 1.12T + 2T^{2}
3 1+(−1.75+1.75i)T−3iT2 1 + (-1.75 + 1.75i)T - 3iT^{2}
7 1+(−1.72−1.72i)T+7iT2 1 + (-1.72 - 1.72i)T + 7iT^{2}
11 1+(−2.57+2.57i)T−11iT2 1 + (-2.57 + 2.57i)T - 11iT^{2}
13 1−3.64iT−13T2 1 - 3.64iT - 13T^{2}
19 1+2.61iT−19T2 1 + 2.61iT - 19T^{2}
23 1+(−0.993−0.993i)T+23iT2 1 + (-0.993 - 0.993i)T + 23iT^{2}
29 1+(0.601+0.601i)T+29iT2 1 + (0.601 + 0.601i)T + 29iT^{2}
31 1+(6.67+6.67i)T+31iT2 1 + (6.67 + 6.67i)T + 31iT^{2}
37 1+(−7.78+7.78i)T−37iT2 1 + (-7.78 + 7.78i)T - 37iT^{2}
41 1+(6.74−6.74i)T−41iT2 1 + (6.74 - 6.74i)T - 41iT^{2}
43 1+7.47T+43T2 1 + 7.47T + 43T^{2}
47 1−5.42iT−47T2 1 - 5.42iT - 47T^{2}
53 1−12.9T+53T2 1 - 12.9T + 53T^{2}
59 1+1.40iT−59T2 1 + 1.40iT - 59T^{2}
61 1+(−0.804+0.804i)T−61iT2 1 + (-0.804 + 0.804i)T - 61iT^{2}
67 1−2.07iT−67T2 1 - 2.07iT - 67T^{2}
71 1+(−8.69−8.69i)T+71iT2 1 + (-8.69 - 8.69i)T + 71iT^{2}
73 1+(1.04−1.04i)T−73iT2 1 + (1.04 - 1.04i)T - 73iT^{2}
79 1+(6.34−6.34i)T−79iT2 1 + (6.34 - 6.34i)T - 79iT^{2}
83 1+2.52T+83T2 1 + 2.52T + 83T^{2}
89 1−1.66T+89T2 1 - 1.66T + 89T^{2}
97 1+(8.67−8.67i)T−97iT2 1 + (8.67 - 8.67i)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

−11.19401428807679805022700740138, −9.594061649081853944558351596640, −9.075924835090840917970591137403, −8.392348322412645083146524810534, −7.65093500335397419028579149146, −6.78250995675399693756444793003, −5.33215637939031244556350617931, −3.91189440311182328895623022895, −2.36863978652721378554944774354, −1.21466502385630849746773851360, 1.52469581997684026580208538731, 3.46295834246923675864236430135, 4.25305704806141476845566387557, 5.23725311522524925381536554277, 7.10846361215202124584124110375, 8.093679424519897464697996450570, 8.581017875013666177869256372232, 9.549152981250602679063676154243, 10.23476905696706962649722815907, 10.66233092643000456334175405511

Graph of the ZZ-function along the critical line