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Properties

Label 2-425-17.9-c1-0-4
Degree 22
Conductor 425425
Sign 0.995−0.0932i0.995 - 0.0932i
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  + (1.09 − 1.09i)2-s + (−2.77 − 1.15i)3-s − 0.419i·4-s + (−4.32 + 1.79i)6-s + (1.32 + 3.19i)7-s + (1.73 + 1.73i)8-s + (4.27 + 4.27i)9-s + (−3.92 + 1.62i)11-s + (−0.483 + 1.16i)12-s + 0.127i·13-s + (4.96 + 2.05i)14-s + 4.66·16-s + (4.11 − 0.193i)17-s + 9.40·18-s + (1.81 − 1.81i)19-s + ⋯
L(s)  = 1  + (0.777 − 0.777i)2-s + (−1.60 − 0.664i)3-s − 0.209i·4-s + (−1.76 + 0.731i)6-s + (0.499 + 1.20i)7-s + (0.614 + 0.614i)8-s + (1.42 + 1.42i)9-s + (−1.18 + 0.489i)11-s + (−0.139 + 0.336i)12-s + 0.0353i·13-s + (1.32 + 0.549i)14-s + 1.16·16-s + (0.998 − 0.0468i)17-s + 2.21·18-s + (0.417 − 0.417i)19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((0.995−0.0932i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0932i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((0.995−0.0932i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0932i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: 0.995−0.0932i0.995 - 0.0932i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(26,⋅)\chi_{425} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), 0.995−0.0932i)(2,\ 425,\ (\ :1/2),\ 0.995 - 0.0932i)

Particular Values

L(1)L(1) ≈\approx 1.20385+0.0562755i1.20385 + 0.0562755i
L(12)L(\frac12) ≈\approx 1.20385+0.0562755i1.20385 + 0.0562755i
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+(−4.11+0.193i)T 1 + (-4.11 + 0.193i)T
good2 1+(−1.09+1.09i)T−2iT2 1 + (-1.09 + 1.09i)T - 2iT^{2}
3 1+(2.77+1.15i)T+(2.12+2.12i)T2 1 + (2.77 + 1.15i)T + (2.12 + 2.12i)T^{2}
7 1+(−1.32−3.19i)T+(−4.94+4.94i)T2 1 + (-1.32 - 3.19i)T + (-4.94 + 4.94i)T^{2}
11 1+(3.92−1.62i)T+(7.77−7.77i)T2 1 + (3.92 - 1.62i)T + (7.77 - 7.77i)T^{2}
13 1−0.127iT−13T2 1 - 0.127iT - 13T^{2}
19 1+(−1.81+1.81i)T−19iT2 1 + (-1.81 + 1.81i)T - 19iT^{2}
23 1+(3.00−1.24i)T+(16.2−16.2i)T2 1 + (3.00 - 1.24i)T + (16.2 - 16.2i)T^{2}
29 1+(1.87−4.53i)T+(−20.5−20.5i)T2 1 + (1.87 - 4.53i)T + (-20.5 - 20.5i)T^{2}
31 1+(−4.95−2.05i)T+(21.9+21.9i)T2 1 + (-4.95 - 2.05i)T + (21.9 + 21.9i)T^{2}
37 1+(−1.63−0.677i)T+(26.1+26.1i)T2 1 + (-1.63 - 0.677i)T + (26.1 + 26.1i)T^{2}
41 1+(−3.85−9.29i)T+(−28.9+28.9i)T2 1 + (-3.85 - 9.29i)T + (-28.9 + 28.9i)T^{2}
43 1+(1.79+1.79i)T+43iT2 1 + (1.79 + 1.79i)T + 43iT^{2}
47 1−4.59iT−47T2 1 - 4.59iT - 47T^{2}
53 1+(1.15−1.15i)T−53iT2 1 + (1.15 - 1.15i)T - 53iT^{2}
59 1+(4.34+4.34i)T+59iT2 1 + (4.34 + 4.34i)T + 59iT^{2}
61 1+(1.54+3.73i)T+(−43.1+43.1i)T2 1 + (1.54 + 3.73i)T + (-43.1 + 43.1i)T^{2}
67 1−6.88T+67T2 1 - 6.88T + 67T^{2}
71 1+(6.66+2.76i)T+(50.2+50.2i)T2 1 + (6.66 + 2.76i)T + (50.2 + 50.2i)T^{2}
73 1+(−5.59+13.5i)T+(−51.6−51.6i)T2 1 + (-5.59 + 13.5i)T + (-51.6 - 51.6i)T^{2}
79 1+(4.75−1.97i)T+(55.8−55.8i)T2 1 + (4.75 - 1.97i)T + (55.8 - 55.8i)T^{2}
83 1+(10.2−10.2i)T−83iT2 1 + (10.2 - 10.2i)T - 83iT^{2}
89 1−0.600iT−89T2 1 - 0.600iT - 89T^{2}
97 1+(−2.93+7.09i)T+(−68.5−68.5i)T2 1 + (-2.93 + 7.09i)T + (-68.5 - 68.5i)T^{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

−11.44971952620389337003675759359, −10.82131860114184795124331234783, −9.839659080187629224601340687244, −8.154501890104553171847268895593, −7.44000560504812601792274825655, −6.04911997622779461269710610886, −5.24461382265393835603237419230, −4.75545985596132018919377297920, −2.85998128938122672819820782373, −1.63624094296647318932086455311, 0.791856920807198456671094187236, 3.83224248747553335091928975278, 4.61086329166831663118854708101, 5.51185489174279175088460615606, 6.00834502636766505256026149445, 7.19726231620007273469876434629, 7.929199704415774439867356414224, 10.00859690040254040504358488583, 10.27042832548700614944361770467, 11.07456510179578133721561304093

Graph of the ZZ-function along the critical line