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Properties

Label 2-425-17.13-c1-0-22
Degree 22
Conductor 425425
Sign −0.980−0.195i-0.980 - 0.195i
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  − 2.24i·2-s + (−0.140 − 0.140i)3-s − 3.05·4-s + (−0.314 + 0.314i)6-s + (1.33 − 1.33i)7-s + 2.37i·8-s − 2.96i·9-s + (2.49 − 2.49i)11-s + (0.428 + 0.428i)12-s − 1.27·13-s + (−3.00 − 3.00i)14-s − 0.766·16-s + (−3.68 + 1.85i)17-s − 6.65·18-s + 4.69i·19-s + ⋯
L(s)  = 1  − 1.59i·2-s + (−0.0808 − 0.0808i)3-s − 1.52·4-s + (−0.128 + 0.128i)6-s + (0.505 − 0.505i)7-s + 0.840i·8-s − 0.986i·9-s + (0.752 − 0.752i)11-s + (0.123 + 0.123i)12-s − 0.353·13-s + (−0.804 − 0.804i)14-s − 0.191·16-s + (−0.893 + 0.449i)17-s − 1.56·18-s + 1.07i·19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((−0.980−0.195i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((−0.980−0.195i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: −0.980−0.195i-0.980 - 0.195i
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), −0.980−0.195i)(2,\ 425,\ (\ :1/2),\ -0.980 - 0.195i)

Particular Values

L(1)L(1) ≈\approx 0.117128+1.18470i0.117128 + 1.18470i
L(12)L(\frac12) ≈\approx 0.117128+1.18470i0.117128 + 1.18470i
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+(3.68−1.85i)T 1 + (3.68 - 1.85i)T
good2 1+2.24iT−2T2 1 + 2.24iT - 2T^{2}
3 1+(0.140+0.140i)T+3iT2 1 + (0.140 + 0.140i)T + 3iT^{2}
7 1+(−1.33+1.33i)T−7iT2 1 + (-1.33 + 1.33i)T - 7iT^{2}
11 1+(−2.49+2.49i)T−11iT2 1 + (-2.49 + 2.49i)T - 11iT^{2}
13 1+1.27T+13T2 1 + 1.27T + 13T^{2}
19 1−4.69iT−19T2 1 - 4.69iT - 19T^{2}
23 1+(−0.406+0.406i)T−23iT2 1 + (-0.406 + 0.406i)T - 23iT^{2}
29 1+(3.81+3.81i)T+29iT2 1 + (3.81 + 3.81i)T + 29iT^{2}
31 1+(4.39+4.39i)T+31iT2 1 + (4.39 + 4.39i)T + 31iT^{2}
37 1+(−6.00−6.00i)T+37iT2 1 + (-6.00 - 6.00i)T + 37iT^{2}
41 1+(−4.28+4.28i)T−41iT2 1 + (-4.28 + 4.28i)T - 41iT^{2}
43 1+9.16iT−43T2 1 + 9.16iT - 43T^{2}
47 1−10.7T+47T2 1 - 10.7T + 47T^{2}
53 1+9.90iT−53T2 1 + 9.90iT - 53T^{2}
59 1+3.15iT−59T2 1 + 3.15iT - 59T^{2}
61 1+(−3.63+3.63i)T−61iT2 1 + (-3.63 + 3.63i)T - 61iT^{2}
67 1+0.281T+67T2 1 + 0.281T + 67T^{2}
71 1+(−8.30−8.30i)T+71iT2 1 + (-8.30 - 8.30i)T + 71iT^{2}
73 1+(−6.95−6.95i)T+73iT2 1 + (-6.95 - 6.95i)T + 73iT^{2}
79 1+(11.9−11.9i)T−79iT2 1 + (11.9 - 11.9i)T - 79iT^{2}
83 1−8.51iT−83T2 1 - 8.51iT - 83T^{2}
89 1−16.7T+89T2 1 - 16.7T + 89T^{2}
97 1+(4.18+4.18i)T+97iT2 1 + (4.18 + 4.18i)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

−10.99438954315923072193109146295, −9.936549197144014218885491980253, −9.232856264128688208229206004480, −8.302559525945363417129076976213, −6.93386910444461926109644220431, −5.78535271433831651990511736649, −4.15041568830681226717066443128, −3.69086205948017133142668399181, −2.13296026176283081126478299432, −0.798235789122958250204344552123, 2.26689207800569984707263466951, 4.47618839874085536885677028207, 5.02002627970418897134385322332, 6.07050324052252353786254843697, 7.18239728740339391756197923784, 7.63314406195769170991243916296, 8.908156428993966611439783299321, 9.270733340205235556274746075159, 10.80371933537512139980493607472, 11.56081921728509270509624029356

Graph of the ZZ-function along the critical line