Properties

Label 2-435-87.17-c1-0-37
Degree 22
Conductor 435435
Sign 0.907+0.419i-0.907 + 0.419i
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  + (−0.398 − 0.398i)2-s + (1.60 − 0.654i)3-s − 1.68i·4-s − 5-s + (−0.899 − 0.378i)6-s − 4.31·7-s + (−1.46 + 1.46i)8-s + (2.14 − 2.10i)9-s + (0.398 + 0.398i)10-s + (0.656 + 0.656i)11-s + (−1.10 − 2.69i)12-s − 2.82i·13-s + (1.72 + 1.72i)14-s + (−1.60 + 0.654i)15-s − 2.19·16-s + (−4.74 − 4.74i)17-s + ⋯
L(s)  = 1  + (−0.281 − 0.281i)2-s + (0.925 − 0.378i)3-s − 0.841i·4-s − 0.447·5-s + (−0.367 − 0.154i)6-s − 1.63·7-s + (−0.518 + 0.518i)8-s + (0.714 − 0.700i)9-s + (0.126 + 0.126i)10-s + (0.197 + 0.197i)11-s + (−0.318 − 0.778i)12-s − 0.782i·13-s + (0.459 + 0.459i)14-s + (−0.414 + 0.169i)15-s − 0.548·16-s + (−1.15 − 1.15i)17-s + ⋯

Functional equation

Λ(s)=(435s/2ΓC(s)L(s)=((0.907+0.419i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(435s/2ΓC(s+1/2)L(s)=((0.907+0.419i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    35293 \cdot 5 \cdot 29
Sign: 0.907+0.419i-0.907 + 0.419i
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), 0.907+0.419i)(2,\ 435,\ (\ :1/2),\ -0.907 + 0.419i)

Particular Values

L(1)L(1) \approx 0.2064880.939530i0.206488 - 0.939530i
L(12)L(\frac12) \approx 0.2064880.939530i0.206488 - 0.939530i
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+(1.60+0.654i)T 1 + (-1.60 + 0.654i)T
5 1+T 1 + T
29 1+(1.255.23i)T 1 + (-1.25 - 5.23i)T
good2 1+(0.398+0.398i)T+2iT2 1 + (0.398 + 0.398i)T + 2iT^{2}
7 1+4.31T+7T2 1 + 4.31T + 7T^{2}
11 1+(0.6560.656i)T+11iT2 1 + (-0.656 - 0.656i)T + 11iT^{2}
13 1+2.82iT13T2 1 + 2.82iT - 13T^{2}
17 1+(4.74+4.74i)T+17iT2 1 + (4.74 + 4.74i)T + 17iT^{2}
19 1+(1.21+1.21i)T19iT2 1 + (-1.21 + 1.21i)T - 19iT^{2}
23 1+1.96iT23T2 1 + 1.96iT - 23T^{2}
31 1+(2.91+2.91i)T31iT2 1 + (-2.91 + 2.91i)T - 31iT^{2}
37 1+(3.893.89i)T+37iT2 1 + (-3.89 - 3.89i)T + 37iT^{2}
41 1+(2.912.91i)T41iT2 1 + (2.91 - 2.91i)T - 41iT^{2}
43 1+(4.82+4.82i)T43iT2 1 + (-4.82 + 4.82i)T - 43iT^{2}
47 1+(6.226.22i)T47iT2 1 + (6.22 - 6.22i)T - 47iT^{2}
53 1+7.97iT53T2 1 + 7.97iT - 53T^{2}
59 1+7.91iT59T2 1 + 7.91iT - 59T^{2}
61 1+(2.74+2.74i)T61iT2 1 + (-2.74 + 2.74i)T - 61iT^{2}
67 1+9.91iT67T2 1 + 9.91iT - 67T^{2}
71 15.70T+71T2 1 - 5.70T + 71T^{2}
73 1+(6.986.98i)T+73iT2 1 + (-6.98 - 6.98i)T + 73iT^{2}
79 1+(3.13+3.13i)T79iT2 1 + (-3.13 + 3.13i)T - 79iT^{2}
83 1+13.0iT83T2 1 + 13.0iT - 83T^{2}
89 1+(10.510.5i)T+89iT2 1 + (-10.5 - 10.5i)T + 89iT^{2}
97 1+(8.438.43i)T+97iT2 1 + (-8.43 - 8.43i)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

−10.54274565737729405946614624313, −9.586075396366748543401558962089, −9.264862823963704990858305669188, −8.169441458158182282392680641084, −6.88836444394377207711988958775, −6.39841912757927267486063808855, −4.83333617533309138276278790309, −3.34977790511709480759962802085, −2.48001595111949212245978587177, −0.57653764907458312612450611667, 2.56944487035404553322893166443, 3.64875092992432578470904675151, 4.20555639468303337190024256276, 6.28067519112364642074708171616, 7.01229647705078325842356520437, 8.009626494648711770495081116960, 8.870306130735585470384664323679, 9.413949458518641815189666863169, 10.35756335658400694111719373736, 11.59384941028493842808588311835

Graph of the ZZ-function along the critical line