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Properties

Label 2-425-85.4-c1-0-5
Degree 22
Conductor 425425
Sign 0.564+0.825i0.564 + 0.825i
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  − 1.21·2-s + (−2.23 − 2.23i)3-s − 0.520·4-s + (2.72 + 2.72i)6-s + (−0.679 + 0.679i)7-s + 3.06·8-s + 7.02i·9-s + (2.22 + 2.22i)11-s + (1.16 + 1.16i)12-s + 2.02i·13-s + (0.827 − 0.827i)14-s − 2.68·16-s + (2.07 − 3.56i)17-s − 8.54i·18-s − 5.28i·19-s + ⋯
L(s)  = 1  − 0.860·2-s + (−1.29 − 1.29i)3-s − 0.260·4-s + (1.11 + 1.11i)6-s + (−0.256 + 0.256i)7-s + 1.08·8-s + 2.34i·9-s + (0.669 + 0.669i)11-s + (0.336 + 0.336i)12-s + 0.561i·13-s + (0.221 − 0.221i)14-s − 0.672·16-s + (0.503 − 0.864i)17-s − 2.01i·18-s − 1.21i·19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((0.564+0.825i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((0.564+0.825i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: 0.564+0.825i0.564 + 0.825i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(174,⋅)\chi_{425} (174, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), 0.564+0.825i)(2,\ 425,\ (\ :1/2),\ 0.564 + 0.825i)

Particular Values

L(1)L(1) ≈\approx 0.401127−0.211733i0.401127 - 0.211733i
L(12)L(\frac12) ≈\approx 0.401127−0.211733i0.401127 - 0.211733i
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+(−2.07+3.56i)T 1 + (-2.07 + 3.56i)T
good2 1+1.21T+2T2 1 + 1.21T + 2T^{2}
3 1+(2.23+2.23i)T+3iT2 1 + (2.23 + 2.23i)T + 3iT^{2}
7 1+(0.679−0.679i)T−7iT2 1 + (0.679 - 0.679i)T - 7iT^{2}
11 1+(−2.22−2.22i)T+11iT2 1 + (-2.22 - 2.22i)T + 11iT^{2}
13 1−2.02iT−13T2 1 - 2.02iT - 13T^{2}
19 1+5.28iT−19T2 1 + 5.28iT - 19T^{2}
23 1+(6.01−6.01i)T−23iT2 1 + (6.01 - 6.01i)T - 23iT^{2}
29 1+(−0.857+0.857i)T−29iT2 1 + (-0.857 + 0.857i)T - 29iT^{2}
31 1+(−3.97+3.97i)T−31iT2 1 + (-3.97 + 3.97i)T - 31iT^{2}
37 1+(−5.84−5.84i)T+37iT2 1 + (-5.84 - 5.84i)T + 37iT^{2}
41 1+(1.04+1.04i)T+41iT2 1 + (1.04 + 1.04i)T + 41iT^{2}
43 1−7.01T+43T2 1 - 7.01T + 43T^{2}
47 1+10.9iT−47T2 1 + 10.9iT - 47T^{2}
53 1−5.24T+53T2 1 - 5.24T + 53T^{2}
59 1+13.8iT−59T2 1 + 13.8iT - 59T^{2}
61 1+(−2.70−2.70i)T+61iT2 1 + (-2.70 - 2.70i)T + 61iT^{2}
67 1−2.37iT−67T2 1 - 2.37iT - 67T^{2}
71 1+(−2.82+2.82i)T−71iT2 1 + (-2.82 + 2.82i)T - 71iT^{2}
73 1+(−5.51−5.51i)T+73iT2 1 + (-5.51 - 5.51i)T + 73iT^{2}
79 1+(−4.74−4.74i)T+79iT2 1 + (-4.74 - 4.74i)T + 79iT^{2}
83 1−0.171T+83T2 1 - 0.171T + 83T^{2}
89 1+1.32T+89T2 1 + 1.32T + 89T^{2}
97 1+(1.33+1.33i)T+97iT2 1 + (1.33 + 1.33i)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

−11.31095567578344952660711356979, −9.997318007073115199336794291048, −9.367170984332468593720853867386, −8.096627566742903106725562401401, −7.28924699606629992974089467739, −6.59978991715011192285217375435, −5.48475760931231231201584621079, −4.44085179156479033080725969491, −2.05209085989288399509527379781, −0.75555773254394704378902217673, 0.821562154540774647364822320444, 3.72844424417423250187669153902, 4.39754983470426262002684093203, 5.68446300726818969773868059476, 6.33829078487114492843962242509, 7.917616023563246987281411825042, 8.838559803524822770364303280112, 9.786787447725747587478663728173, 10.38709540108969963309839691362, 10.80773339914138762663958305524

Graph of the ZZ-function along the critical line