Properties

Label 2-755-5.4-c1-0-49
Degree 22
Conductor 755755
Sign 0.936+0.350i0.936 + 0.350i
Analytic cond. 6.028706.02870
Root an. cond. 2.455342.45534
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  + 1.84i·2-s + 2.98i·3-s − 1.39·4-s + (−2.09 − 0.784i)5-s − 5.49·6-s − 5.05i·7-s + 1.11i·8-s − 5.90·9-s + (1.44 − 3.85i)10-s − 3.23·11-s − 4.15i·12-s − 2.63i·13-s + 9.31·14-s + (2.34 − 6.25i)15-s − 4.84·16-s − 0.962i·17-s + ⋯
L(s)  = 1  + 1.30i·2-s + 1.72i·3-s − 0.696·4-s + (−0.936 − 0.350i)5-s − 2.24·6-s − 1.91i·7-s + 0.395i·8-s − 1.96·9-s + (0.456 − 1.21i)10-s − 0.974·11-s − 1.19i·12-s − 0.731i·13-s + 2.48·14-s + (0.604 − 1.61i)15-s − 1.21·16-s − 0.233i·17-s + ⋯

Functional equation

Λ(s)=(755s/2ΓC(s)L(s)=((0.936+0.350i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(755s/2ΓC(s+1/2)L(s)=((0.936+0.350i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 755755    =    51515 \cdot 151
Sign: 0.936+0.350i0.936 + 0.350i
Analytic conductor: 6.028706.02870
Root analytic conductor: 2.455342.45534
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ755(454,)\chi_{755} (454, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 755, ( :1/2), 0.936+0.350i)(2,\ 755,\ (\ :1/2),\ 0.936 + 0.350i)

Particular Values

L(1)L(1) \approx 0.1627920.0294775i0.162792 - 0.0294775i
L(12)L(\frac12) \approx 0.1627920.0294775i0.162792 - 0.0294775i
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)
bad5 1+(2.09+0.784i)T 1 + (2.09 + 0.784i)T
151 1+T 1 + T
good2 11.84iT2T2 1 - 1.84iT - 2T^{2}
3 12.98iT3T2 1 - 2.98iT - 3T^{2}
7 1+5.05iT7T2 1 + 5.05iT - 7T^{2}
11 1+3.23T+11T2 1 + 3.23T + 11T^{2}
13 1+2.63iT13T2 1 + 2.63iT - 13T^{2}
17 1+0.962iT17T2 1 + 0.962iT - 17T^{2}
19 12.66T+19T2 1 - 2.66T + 19T^{2}
23 15.80iT23T2 1 - 5.80iT - 23T^{2}
29 1+2.13T+29T2 1 + 2.13T + 29T^{2}
31 1+6.68T+31T2 1 + 6.68T + 31T^{2}
37 1+7.66iT37T2 1 + 7.66iT - 37T^{2}
41 1+9.30T+41T2 1 + 9.30T + 41T^{2}
43 1+5.98iT43T2 1 + 5.98iT - 43T^{2}
47 1+7.77iT47T2 1 + 7.77iT - 47T^{2}
53 14.36iT53T2 1 - 4.36iT - 53T^{2}
59 1+11.8T+59T2 1 + 11.8T + 59T^{2}
61 16.78T+61T2 1 - 6.78T + 61T^{2}
67 12.79iT67T2 1 - 2.79iT - 67T^{2}
71 114.5T+71T2 1 - 14.5T + 71T^{2}
73 111.7iT73T2 1 - 11.7iT - 73T^{2}
79 1+4.34T+79T2 1 + 4.34T + 79T^{2}
83 1+8.54iT83T2 1 + 8.54iT - 83T^{2}
89 1+12.4T+89T2 1 + 12.4T + 89T^{2}
97 16.10iT97T2 1 - 6.10iT - 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

−10.30333875360952798781709472817, −9.387615557818995146439873288163, −8.389562760699132963562659164642, −7.58670052591356537836982618873, −7.15820736795649262249584893208, −5.43910795319931736123141901347, −5.07220722375609586393717253738, −4.00345946960510397694997152680, −3.40940501466438605523173346494, −0.080161995680484824332040155426, 1.69944019164978905890006308332, 2.54280845588989417291192684690, 3.19229303221721661790313441936, 4.93967457959929073612613138867, 6.18306263988617094061904019054, 6.91057718280963438513554612564, 7.976709364191828230065297476507, 8.568841900875826006637661958777, 9.537959346595162183099081700923, 10.88502648720345787207292537336

Graph of the ZZ-function along the critical line