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Properties

Label 2-825-11.4-c1-0-29
Degree 22
Conductor 825825
Sign 0.785+0.619i0.785 + 0.619i
Analytic cond. 6.587656.58765
Root an. cond. 2.566642.56664
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  + (1.54 + 1.12i)2-s + (0.309 − 0.951i)3-s + (0.506 + 1.55i)4-s + (1.54 − 1.12i)6-s + (−1.37 − 4.23i)7-s + (0.213 − 0.656i)8-s + (−0.809 − 0.587i)9-s + (−1.22 + 3.08i)11-s + 1.63·12-s + (−0.432 − 0.313i)13-s + (2.62 − 8.08i)14-s + (3.71 − 2.69i)16-s + (3.67 − 2.67i)17-s + (−0.589 − 1.81i)18-s + (0.594 − 1.83i)19-s + ⋯
L(s)  = 1  + (1.09 + 0.792i)2-s + (0.178 − 0.549i)3-s + (0.253 + 0.779i)4-s + (0.629 − 0.457i)6-s + (−0.520 − 1.60i)7-s + (0.0753 − 0.231i)8-s + (−0.269 − 0.195i)9-s + (−0.368 + 0.929i)11-s + 0.472·12-s + (−0.119 − 0.0870i)13-s + (0.702 − 2.16i)14-s + (0.928 − 0.674i)16-s + (0.891 − 0.647i)17-s + (−0.138 − 0.427i)18-s + (0.136 − 0.419i)19-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=((0.785+0.619i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s+1/2) L(s)=((0.785+0.619i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: 0.785+0.619i0.785 + 0.619i
Analytic conductor: 6.587656.58765
Root analytic conductor: 2.566642.56664
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ825(301,⋅)\chi_{825} (301, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 825, ( :1/2), 0.785+0.619i)(2,\ 825,\ (\ :1/2),\ 0.785 + 0.619i)

Particular Values

L(1)L(1) ≈\approx 2.48997−0.863364i2.48997 - 0.863364i
L(12)L(\frac12) ≈\approx 2.48997−0.863364i2.48997 - 0.863364i
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)
bad3 1+(−0.309+0.951i)T 1 + (-0.309 + 0.951i)T
5 1 1
11 1+(1.22−3.08i)T 1 + (1.22 - 3.08i)T
good2 1+(−1.54−1.12i)T+(0.618+1.90i)T2 1 + (-1.54 - 1.12i)T + (0.618 + 1.90i)T^{2}
7 1+(1.37+4.23i)T+(−5.66+4.11i)T2 1 + (1.37 + 4.23i)T + (-5.66 + 4.11i)T^{2}
13 1+(0.432+0.313i)T+(4.01+12.3i)T2 1 + (0.432 + 0.313i)T + (4.01 + 12.3i)T^{2}
17 1+(−3.67+2.67i)T+(5.25−16.1i)T2 1 + (-3.67 + 2.67i)T + (5.25 - 16.1i)T^{2}
19 1+(−0.594+1.83i)T+(−15.3−11.1i)T2 1 + (-0.594 + 1.83i)T + (-15.3 - 11.1i)T^{2}
23 1−5.18T+23T2 1 - 5.18T + 23T^{2}
29 1+(1.46+4.52i)T+(−23.4+17.0i)T2 1 + (1.46 + 4.52i)T + (-23.4 + 17.0i)T^{2}
31 1+(2.82+2.05i)T+(9.57+29.4i)T2 1 + (2.82 + 2.05i)T + (9.57 + 29.4i)T^{2}
37 1+(−0.162−0.501i)T+(−29.9+21.7i)T2 1 + (-0.162 - 0.501i)T + (-29.9 + 21.7i)T^{2}
41 1+(−1.45+4.46i)T+(−33.1−24.0i)T2 1 + (-1.45 + 4.46i)T + (-33.1 - 24.0i)T^{2}
43 1+3.02T+43T2 1 + 3.02T + 43T^{2}
47 1+(3.70−11.4i)T+(−38.0−27.6i)T2 1 + (3.70 - 11.4i)T + (-38.0 - 27.6i)T^{2}
53 1+(−4.06−2.95i)T+(16.3+50.4i)T2 1 + (-4.06 - 2.95i)T + (16.3 + 50.4i)T^{2}
59 1+(−4.21−12.9i)T+(−47.7+34.6i)T2 1 + (-4.21 - 12.9i)T + (-47.7 + 34.6i)T^{2}
61 1+(9.45−6.87i)T+(18.8−58.0i)T2 1 + (9.45 - 6.87i)T + (18.8 - 58.0i)T^{2}
67 1−11.5T+67T2 1 - 11.5T + 67T^{2}
71 1+(2.72−1.97i)T+(21.9−67.5i)T2 1 + (2.72 - 1.97i)T + (21.9 - 67.5i)T^{2}
73 1+(−0.200−0.618i)T+(−59.0+42.9i)T2 1 + (-0.200 - 0.618i)T + (-59.0 + 42.9i)T^{2}
79 1+(−14.3−10.4i)T+(24.4+75.1i)T2 1 + (-14.3 - 10.4i)T + (24.4 + 75.1i)T^{2}
83 1+(−3.43+2.49i)T+(25.6−78.9i)T2 1 + (-3.43 + 2.49i)T + (25.6 - 78.9i)T^{2}
89 1+8.24T+89T2 1 + 8.24T + 89T^{2}
97 1+(−0.920−0.669i)T+(29.9+92.2i)T2 1 + (-0.920 - 0.669i)T + (29.9 + 92.2i)T^{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

−10.05873521675124567247887110872, −9.424918037239332634385898802485, −7.81184605133814349977864536785, −7.27478513732231758686029221285, −6.85475307351203690383291792163, −5.75210786045235340721528589514, −4.76688158386599733774350994885, −3.94121324331542721503344808624, −2.89625105015649457369614234096, −0.938897240217945778556641165752, 2.00663805794076393969027779575, 3.17311405871891537273421801445, 3.50002210693355203565624548456, 5.11226707709350852371515094662, 5.42809893107952375028084168071, 6.39765905007425267409473008307, 8.060367526572713279284203215939, 8.711596110935513618584262962893, 9.588456614803247860761139674961, 10.54986245203523800555827580930

Graph of the ZZ-function along the critical line