Properties

Label 2-435-87.17-c1-0-32
Degree 22
Conductor 435435
Sign 0.0254+0.999i0.0254 + 0.999i
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  + (0.0236 + 0.0236i)2-s + (1.39 − 1.03i)3-s − 1.99i·4-s − 5-s + (0.0572 + 0.00855i)6-s + 3.42·7-s + (0.0944 − 0.0944i)8-s + (0.877 − 2.86i)9-s + (−0.0236 − 0.0236i)10-s + (−4.10 − 4.10i)11-s + (−2.05 − 2.78i)12-s + 3.19i·13-s + (0.0809 + 0.0809i)14-s + (−1.39 + 1.03i)15-s − 3.99·16-s + (4.08 + 4.08i)17-s + ⋯
L(s)  = 1  + (0.0167 + 0.0167i)2-s + (0.803 − 0.594i)3-s − 0.999i·4-s − 0.447·5-s + (0.0233 + 0.00349i)6-s + 1.29·7-s + (0.0334 − 0.0334i)8-s + (0.292 − 0.956i)9-s + (−0.00747 − 0.00747i)10-s + (−1.23 − 1.23i)11-s + (−0.594 − 0.803i)12-s + 0.886i·13-s + (0.0216 + 0.0216i)14-s + (−0.359 + 0.266i)15-s − 0.998·16-s + (0.991 + 0.991i)17-s + ⋯

Functional equation

Λ(s)=(435s/2ΓC(s)L(s)=((0.0254+0.999i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0254 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(435s/2ΓC(s+1/2)L(s)=((0.0254+0.999i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0254 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    35293 \cdot 5 \cdot 29
Sign: 0.0254+0.999i0.0254 + 0.999i
Analytic conductor: 3.473493.47349
Root analytic conductor: 1.863731.86373
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ435(191,)\chi_{435} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 435, ( :1/2), 0.0254+0.999i)(2,\ 435,\ (\ :1/2),\ 0.0254 + 0.999i)

Particular Values

L(1)L(1) \approx 1.279941.24771i1.27994 - 1.24771i
L(12)L(\frac12) \approx 1.279941.24771i1.27994 - 1.24771i
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+(1.39+1.03i)T 1 + (-1.39 + 1.03i)T
5 1+T 1 + T
29 1+(4.912.20i)T 1 + (-4.91 - 2.20i)T
good2 1+(0.02360.0236i)T+2iT2 1 + (-0.0236 - 0.0236i)T + 2iT^{2}
7 13.42T+7T2 1 - 3.42T + 7T^{2}
11 1+(4.10+4.10i)T+11iT2 1 + (4.10 + 4.10i)T + 11iT^{2}
13 13.19iT13T2 1 - 3.19iT - 13T^{2}
17 1+(4.084.08i)T+17iT2 1 + (-4.08 - 4.08i)T + 17iT^{2}
19 1+(1.511.51i)T19iT2 1 + (1.51 - 1.51i)T - 19iT^{2}
23 1+3.49iT23T2 1 + 3.49iT - 23T^{2}
31 1+(0.8960.896i)T31iT2 1 + (0.896 - 0.896i)T - 31iT^{2}
37 1+(1.53+1.53i)T+37iT2 1 + (1.53 + 1.53i)T + 37iT^{2}
41 1+(4.78+4.78i)T41iT2 1 + (-4.78 + 4.78i)T - 41iT^{2}
43 1+(2.792.79i)T43iT2 1 + (2.79 - 2.79i)T - 43iT^{2}
47 1+(3.94+3.94i)T47iT2 1 + (-3.94 + 3.94i)T - 47iT^{2}
53 1+0.219iT53T2 1 + 0.219iT - 53T^{2}
59 114.4iT59T2 1 - 14.4iT - 59T^{2}
61 1+(7.78+7.78i)T61iT2 1 + (-7.78 + 7.78i)T - 61iT^{2}
67 111.2iT67T2 1 - 11.2iT - 67T^{2}
71 1+9.21T+71T2 1 + 9.21T + 71T^{2}
73 1+(10.710.7i)T+73iT2 1 + (-10.7 - 10.7i)T + 73iT^{2}
79 1+(1.49+1.49i)T79iT2 1 + (-1.49 + 1.49i)T - 79iT^{2}
83 11.76iT83T2 1 - 1.76iT - 83T^{2}
89 1+(6.116.11i)T+89iT2 1 + (-6.11 - 6.11i)T + 89iT^{2}
97 1+(5.10+5.10i)T+97iT2 1 + (5.10 + 5.10i)T + 97iT^{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

−10.84535247001714738357918066167, −10.18807492218347895753598785646, −8.714103004388807454009552751520, −8.342807998924989513783188531996, −7.38773267118106232231392313357, −6.19405814086199974213811061445, −5.20788547953232194174530409715, −3.96911021512214635763628011280, −2.41151709122145508756592823456, −1.15469390156596609730449845957, 2.30755848805507095047063974169, 3.28477833901139002699005209096, 4.60381246969465014438343304023, 5.08047621668424795019586620121, 7.39547770995674206698928546158, 7.78083408612144466835026009229, 8.341958926279087772042391396612, 9.523260654606446791907346061436, 10.45011887061128635348232343316, 11.34011938489439266124600458529

Graph of the ZZ-function along the critical line