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Properties

Label 2-425-5.4-c1-0-11
Degree 22
Conductor 425425
Sign 0.447−0.894i0.447 - 0.894i
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  + 2.60i·2-s − 1.18i·3-s − 4.77·4-s + 3.07·6-s − 3.53i·7-s − 7.21i·8-s + 1.60·9-s + 2.94·11-s + 5.64i·12-s + 4.01i·13-s + 9.20·14-s + 9.23·16-s + i·17-s + 4.17i·18-s + 6.97·19-s + ⋯
L(s)  = 1  + 1.84i·2-s − 0.682i·3-s − 2.38·4-s + 1.25·6-s − 1.33i·7-s − 2.55i·8-s + 0.534·9-s + 0.888·11-s + 1.62i·12-s + 1.11i·13-s + 2.45·14-s + 2.30·16-s + 0.242i·17-s + 0.982i·18-s + 1.60·19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(324,⋅)\chi_{425} (324, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), 0.447−0.894i)(2,\ 425,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.11133+0.686841i1.11133 + 0.686841i
L(12)L(\frac12) ≈\approx 1.11133+0.686841i1.11133 + 0.686841i
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−iT 1 - iT
good2 1−2.60iT−2T2 1 - 2.60iT - 2T^{2}
3 1+1.18iT−3T2 1 + 1.18iT - 3T^{2}
7 1+3.53iT−7T2 1 + 3.53iT - 7T^{2}
11 1−2.94T+11T2 1 - 2.94T + 11T^{2}
13 1−4.01iT−13T2 1 - 4.01iT - 13T^{2}
19 1−6.97T+19T2 1 - 6.97T + 19T^{2}
23 1+6.12iT−23T2 1 + 6.12iT - 23T^{2}
29 1+5.30T+29T2 1 + 5.30T + 29T^{2}
31 1−6.49T+31T2 1 - 6.49T + 31T^{2}
37 1+3.43iT−37T2 1 + 3.43iT - 37T^{2}
41 1−4.61T+41T2 1 - 4.61T + 41T^{2}
43 1−10.2iT−43T2 1 - 10.2iT - 43T^{2}
47 1+3.67iT−47T2 1 + 3.67iT - 47T^{2}
53 1+6.77iT−53T2 1 + 6.77iT - 53T^{2}
59 1+9.92T+59T2 1 + 9.92T + 59T^{2}
61 1+2.36T+61T2 1 + 2.36T + 61T^{2}
67 1−9.56iT−67T2 1 - 9.56iT - 67T^{2}
71 1−5.51T+71T2 1 - 5.51T + 71T^{2}
73 1−2.00iT−73T2 1 - 2.00iT - 73T^{2}
79 1+10.5T+79T2 1 + 10.5T + 79T^{2}
83 1−9.07iT−83T2 1 - 9.07iT - 83T^{2}
89 1+2.63T+89T2 1 + 2.63T + 89T^{2}
97 1+5.86iT−97T2 1 + 5.86iT - 97T^{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

−11.43210721596858629466466408306, −9.990754210258589083741113749453, −9.298604174750196093874098908080, −8.158411995363067398596717412508, −7.31557843041070624485363852699, −6.87287151942463214187884599847, −6.11819981016611825560801935751, −4.62538315514944369514576233336, −3.96962045639960560314002283185, −1.08643592367414973503254490501, 1.40673212829639312385410396829, 2.88374759307038340463624944815, 3.67483966662507325573209796404, 4.89797794904517268884112418508, 5.70037057967726403412558192624, 7.68284614012553487898814891924, 9.022280041372009947997764724996, 9.420240838368438067114765426022, 10.12657452675738637520919582224, 11.09931396819239483260908206238

Graph of the ZZ-function along the critical line