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Properties

Label 2-425-17.13-c1-0-6
Degree 22
Conductor 425425
Sign 0.990+0.135i0.990 + 0.135i
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  + 1.21i·2-s + (−2.23 − 2.23i)3-s + 0.520·4-s + (2.72 − 2.72i)6-s + (−0.679 + 0.679i)7-s + 3.06i·8-s + 7.02i·9-s + (2.22 − 2.22i)11-s + (−1.16 − 1.16i)12-s + 2.02·13-s + (−0.827 − 0.827i)14-s − 2.68·16-s + (3.56 − 2.07i)17-s − 8.54·18-s − 5.28i·19-s + ⋯
L(s)  = 1  + 0.860i·2-s + (−1.29 − 1.29i)3-s + 0.260·4-s + (1.11 − 1.11i)6-s + (−0.256 + 0.256i)7-s + 1.08i·8-s + 2.34i·9-s + (0.669 − 0.669i)11-s + (−0.336 − 0.336i)12-s + 0.561·13-s + (−0.221 − 0.221i)14-s − 0.672·16-s + (0.864 − 0.503i)17-s − 2.01·18-s − 1.21i·19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((0.990+0.135i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((0.990+0.135i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: 0.990+0.135i0.990 + 0.135i
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), 0.990+0.135i)(2,\ 425,\ (\ :1/2),\ 0.990 + 0.135i)

Particular Values

L(1)L(1) ≈\approx 1.07651−0.0732076i1.07651 - 0.0732076i
L(12)L(\frac12) ≈\approx 1.07651−0.0732076i1.07651 - 0.0732076i
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+(−3.56+2.07i)T 1 + (-3.56 + 2.07i)T
good2 1−1.21iT−2T2 1 - 1.21iT - 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 1−2.02T+13T2 1 - 2.02T + 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 1−7.01iT−43T2 1 - 7.01iT - 43T^{2}
47 1−10.9T+47T2 1 - 10.9T + 47T^{2}
53 1−5.24iT−53T2 1 - 5.24iT - 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+2.37T+67T2 1 + 2.37T + 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 1−0.171iT−83T2 1 - 0.171iT - 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

−11.20399256800255559715818917298, −10.77983796413381865192743287770, −9.005557921134201525655435353569, −8.056261613932580050007359937410, −7.01362444847559464363508618839, −6.59137506060587188829929547755, −5.78691250873476577143970032453, −4.96870358908261533122289882877, −2.69414271355665950304829884478, −1.00925039481378464120429857240, 1.26845639353507547550243170261, 3.48875523758687329252596456282, 4.01334156760164855632224732294, 5.36932314305888699493232925893, 6.25334180643449513207911414467, 7.22567147569096910972477972342, 9.010522691618971728277369899766, 10.00398028635560156580430433217, 10.26724863549099490425396247254, 11.14626798210802396872530386710

Graph of the ZZ-function along the critical line