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Properties

Label 2-425-17.13-c1-0-4
Degree 22
Conductor 425425
Sign −0.944−0.327i-0.944 - 0.327i
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  + 1.68i·2-s + (1.27 + 1.27i)3-s − 0.830·4-s + (−2.14 + 2.14i)6-s + (−1.92 + 1.92i)7-s + 1.96i·8-s + 0.249i·9-s + (−0.0173 + 0.0173i)11-s + (−1.05 − 1.05i)12-s + 3.62·13-s + (−3.24 − 3.24i)14-s − 4.97·16-s + (−3.90 + 1.33i)17-s − 0.419·18-s + 0.603i·19-s + ⋯
L(s)  = 1  + 1.18i·2-s + (0.735 + 0.735i)3-s − 0.415·4-s + (−0.875 + 0.875i)6-s + (−0.728 + 0.728i)7-s + 0.695i·8-s + 0.0830i·9-s + (−0.00524 + 0.00524i)11-s + (−0.305 − 0.305i)12-s + 1.00·13-s + (−0.866 − 0.866i)14-s − 1.24·16-s + (−0.946 + 0.323i)17-s − 0.0987·18-s + 0.138i·19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((−0.944−0.327i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((−0.944−0.327i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: −0.944−0.327i-0.944 - 0.327i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(251,⋅)\chi_{425} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), −0.944−0.327i)(2,\ 425,\ (\ :1/2),\ -0.944 - 0.327i)

Particular Values

L(1)L(1) ≈\approx 0.282613+1.67750i0.282613 + 1.67750i
L(12)L(\frac12) ≈\approx 0.282613+1.67750i0.282613 + 1.67750i
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+(3.90−1.33i)T 1 + (3.90 - 1.33i)T
good2 1−1.68iT−2T2 1 - 1.68iT - 2T^{2}
3 1+(−1.27−1.27i)T+3iT2 1 + (-1.27 - 1.27i)T + 3iT^{2}
7 1+(1.92−1.92i)T−7iT2 1 + (1.92 - 1.92i)T - 7iT^{2}
11 1+(0.0173−0.0173i)T−11iT2 1 + (0.0173 - 0.0173i)T - 11iT^{2}
13 1−3.62T+13T2 1 - 3.62T + 13T^{2}
19 1−0.603iT−19T2 1 - 0.603iT - 19T^{2}
23 1+(−1.94+1.94i)T−23iT2 1 + (-1.94 + 1.94i)T - 23iT^{2}
29 1+(−3.01−3.01i)T+29iT2 1 + (-3.01 - 3.01i)T + 29iT^{2}
31 1+(0.422+0.422i)T+31iT2 1 + (0.422 + 0.422i)T + 31iT^{2}
37 1+(−7.50−7.50i)T+37iT2 1 + (-7.50 - 7.50i)T + 37iT^{2}
41 1+(−5.07+5.07i)T−41iT2 1 + (-5.07 + 5.07i)T - 41iT^{2}
43 1+12.0iT−43T2 1 + 12.0iT - 43T^{2}
47 1+11.0T+47T2 1 + 11.0T + 47T^{2}
53 1−2.05iT−53T2 1 - 2.05iT - 53T^{2}
59 1+0.926iT−59T2 1 + 0.926iT - 59T^{2}
61 1+(−8.41+8.41i)T−61iT2 1 + (-8.41 + 8.41i)T - 61iT^{2}
67 1−5.79T+67T2 1 - 5.79T + 67T^{2}
71 1+(3.84+3.84i)T+71iT2 1 + (3.84 + 3.84i)T + 71iT^{2}
73 1+(−2.88−2.88i)T+73iT2 1 + (-2.88 - 2.88i)T + 73iT^{2}
79 1+(−9.68+9.68i)T−79iT2 1 + (-9.68 + 9.68i)T - 79iT^{2}
83 1−12.1iT−83T2 1 - 12.1iT - 83T^{2}
89 1+7.11T+89T2 1 + 7.11T + 89T^{2}
97 1+(−5.01−5.01i)T+97iT2 1 + (-5.01 - 5.01i)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

−11.48095734878408961810680718652, −10.50237611821564192309745886471, −9.346860442075471723268311248827, −8.763000572957161269038009777325, −8.115780675191957449781313634504, −6.68019588733635222225959363341, −6.19465019792869481268167325853, −4.97312596111349686732352829167, −3.72802802400603340031608880037, −2.53470155161430605108931351630, 1.07038090426878260619949525002, 2.40725948490643318158253827361, 3.32409029753353059268995043521, 4.39751252830110353611587318084, 6.31818665438502554196585366480, 7.07500064357585994204068076252, 8.100626074128502113965630360794, 9.178963238644997150485438093086, 9.942771538961050831882192607802, 10.99954456983189083452633412283

Graph of the ZZ-function along the critical line