Properties

Label 2-435-145.144-c1-0-5
Degree 22
Conductor 435435
Sign 0.4480.893i0.448 - 0.893i
Analytic cond. 3.473493.47349
Root an. cond. 1.863731.86373
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  − 1.97·2-s − 3-s + 1.89·4-s + (1.38 + 1.75i)5-s + 1.97·6-s − 0.520i·7-s + 0.214·8-s + 9-s + (−2.72 − 3.46i)10-s + 2.16i·11-s − 1.89·12-s − 1.53i·13-s + 1.02i·14-s + (−1.38 − 1.75i)15-s − 4.20·16-s + 2.47·17-s + ⋯
L(s)  = 1  − 1.39·2-s − 0.577·3-s + 0.945·4-s + (0.617 + 0.786i)5-s + 0.805·6-s − 0.196i·7-s + 0.0758·8-s + 0.333·9-s + (−0.861 − 1.09i)10-s + 0.652i·11-s − 0.545·12-s − 0.425i·13-s + 0.274i·14-s + (−0.356 − 0.454i)15-s − 1.05·16-s + 0.601·17-s + ⋯

Functional equation

Λ(s)=(435s/2ΓC(s)L(s)=((0.4480.893i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(435s/2ΓC(s+1/2)L(s)=((0.4480.893i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    35293 \cdot 5 \cdot 29
Sign: 0.4480.893i0.448 - 0.893i
Analytic conductor: 3.473493.47349
Root analytic conductor: 1.863731.86373
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ435(289,)\chi_{435} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 435, ( :1/2), 0.4480.893i)(2,\ 435,\ (\ :1/2),\ 0.448 - 0.893i)

Particular Values

L(1)L(1) \approx 0.501384+0.309462i0.501384 + 0.309462i
L(12)L(\frac12) \approx 0.501384+0.309462i0.501384 + 0.309462i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+T 1 + T
5 1+(1.381.75i)T 1 + (-1.38 - 1.75i)T
29 1+(2.294.87i)T 1 + (-2.29 - 4.87i)T
good2 1+1.97T+2T2 1 + 1.97T + 2T^{2}
7 1+0.520iT7T2 1 + 0.520iT - 7T^{2}
11 12.16iT11T2 1 - 2.16iT - 11T^{2}
13 1+1.53iT13T2 1 + 1.53iT - 13T^{2}
17 12.47T+17T2 1 - 2.47T + 17T^{2}
19 1+4.70iT19T2 1 + 4.70iT - 19T^{2}
23 11.46iT23T2 1 - 1.46iT - 23T^{2}
31 14.64iT31T2 1 - 4.64iT - 31T^{2}
37 11.77T+37T2 1 - 1.77T + 37T^{2}
41 19.71iT41T2 1 - 9.71iT - 41T^{2}
43 15.39T+43T2 1 - 5.39T + 43T^{2}
47 13.68T+47T2 1 - 3.68T + 47T^{2}
53 17.74iT53T2 1 - 7.74iT - 53T^{2}
59 1+1.91T+59T2 1 + 1.91T + 59T^{2}
61 17.66iT61T2 1 - 7.66iT - 61T^{2}
67 17.88iT67T2 1 - 7.88iT - 67T^{2}
71 1+9.78T+71T2 1 + 9.78T + 71T^{2}
73 1+4.70T+73T2 1 + 4.70T + 73T^{2}
79 1+12.3iT79T2 1 + 12.3iT - 79T^{2}
83 17.32iT83T2 1 - 7.32iT - 83T^{2}
89 1+7.66iT89T2 1 + 7.66iT - 89T^{2}
97 1+1.64T+97T2 1 + 1.64T + 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−10.84229579715468130348142278173, −10.36244088380221644216095378385, −9.636508696064823728799599702272, −8.786248311154622492071496998094, −7.46136687014584946002846181141, −7.03758658382554884502711087736, −5.88797078196181595215712003270, −4.63372524825557059587573872200, −2.79363568961288372718918188206, −1.28314545827527783757071887383, 0.74035514304204564601057686220, 2.05689704942115635023617649873, 4.19185150448646007936721143228, 5.50199111514738442427427111391, 6.30520536078259477546922755290, 7.60591039136979406012883783845, 8.422400883984321005602303818435, 9.227828649709376807035745704167, 9.963177360252382493408876557808, 10.67305537593214080853953781178

Graph of the ZZ-function along the critical line