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Properties

Label 2-425-17.13-c1-0-1
Degree 22
Conductor 425425
Sign −0.957−0.289i-0.957 - 0.289i
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  + 2.07i·2-s + (−1.78 − 1.78i)3-s − 2.28·4-s + (3.69 − 3.69i)6-s + (0.260 − 0.260i)7-s − 0.595i·8-s + 3.36i·9-s + (−1.76 + 1.76i)11-s + (4.08 + 4.08i)12-s + 4.68·13-s + (0.540 + 0.540i)14-s − 3.34·16-s + (−3.84 + 1.48i)17-s − 6.97·18-s + 7.16i·19-s + ⋯
L(s)  = 1  + 1.46i·2-s + (−1.03 − 1.03i)3-s − 1.14·4-s + (1.50 − 1.50i)6-s + (0.0986 − 0.0986i)7-s − 0.210i·8-s + 1.12i·9-s + (−0.532 + 0.532i)11-s + (1.17 + 1.17i)12-s + 1.30·13-s + (0.144 + 0.144i)14-s − 0.835·16-s + (−0.932 + 0.360i)17-s − 1.64·18-s + 1.64i·19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((−0.957−0.289i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((−0.957−0.289i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: −0.957−0.289i-0.957 - 0.289i
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), −0.957−0.289i)(2,\ 425,\ (\ :1/2),\ -0.957 - 0.289i)

Particular Values

L(1)L(1) ≈\approx 0.0927958+0.626893i0.0927958 + 0.626893i
L(12)L(\frac12) ≈\approx 0.0927958+0.626893i0.0927958 + 0.626893i
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+(3.84−1.48i)T 1 + (3.84 - 1.48i)T
good2 1−2.07iT−2T2 1 - 2.07iT - 2T^{2}
3 1+(1.78+1.78i)T+3iT2 1 + (1.78 + 1.78i)T + 3iT^{2}
7 1+(−0.260+0.260i)T−7iT2 1 + (-0.260 + 0.260i)T - 7iT^{2}
11 1+(1.76−1.76i)T−11iT2 1 + (1.76 - 1.76i)T - 11iT^{2}
13 1−4.68T+13T2 1 - 4.68T + 13T^{2}
19 1−7.16iT−19T2 1 - 7.16iT - 19T^{2}
23 1+(4.73−4.73i)T−23iT2 1 + (4.73 - 4.73i)T - 23iT^{2}
29 1+(4.79+4.79i)T+29iT2 1 + (4.79 + 4.79i)T + 29iT^{2}
31 1+(−3.40−3.40i)T+31iT2 1 + (-3.40 - 3.40i)T + 31iT^{2}
37 1+(−1.37−1.37i)T+37iT2 1 + (-1.37 - 1.37i)T + 37iT^{2}
41 1+(−1.66+1.66i)T−41iT2 1 + (-1.66 + 1.66i)T - 41iT^{2}
43 1−11.7iT−43T2 1 - 11.7iT - 43T^{2}
47 1−1.65T+47T2 1 - 1.65T + 47T^{2}
53 1+6.81iT−53T2 1 + 6.81iT - 53T^{2}
59 1+0.484iT−59T2 1 + 0.484iT - 59T^{2}
61 1+(−4.86+4.86i)T−61iT2 1 + (-4.86 + 4.86i)T - 61iT^{2}
67 1−1.87T+67T2 1 - 1.87T + 67T^{2}
71 1+(−1.21−1.21i)T+71iT2 1 + (-1.21 - 1.21i)T + 71iT^{2}
73 1+(0.202+0.202i)T+73iT2 1 + (0.202 + 0.202i)T + 73iT^{2}
79 1+(−3.80+3.80i)T−79iT2 1 + (-3.80 + 3.80i)T - 79iT^{2}
83 1−9.94iT−83T2 1 - 9.94iT - 83T^{2}
89 1+4.30T+89T2 1 + 4.30T + 89T^{2}
97 1+(9.01+9.01i)T+97iT2 1 + (9.01 + 9.01i)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

−11.59809718327628569063631343246, −10.90858340931274775886810446985, −9.624211600062659587103656690398, −8.184222677648773413021326338861, −7.81689570937550426145412206666, −6.73366644460602954234824735817, −6.07927571486740227209958893553, −5.50578810535654991705719483309, −4.14373628099135465223505056796, −1.72211865247491794086935508072, 0.45878954507378967587357711680, 2.44791485572697754172012076659, 3.79095893092929381277233356797, 4.58804464149939646880023162116, 5.64799129982182478469890799186, 6.76157478397282389371695667056, 8.582855394948275212361298319782, 9.303038830987056504183484595935, 10.34590135733983997542488987920, 10.93719109052559768809169540282

Graph of the ZZ-function along the critical line