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Properties

Label 2-825-11.3-c1-0-34
Degree 22
Conductor 825825
Sign −0.100+0.994i-0.100 + 0.994i
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  + (2.20 − 1.60i)2-s + (0.309 + 0.951i)3-s + (1.67 − 5.16i)4-s + (2.20 + 1.60i)6-s + (−0.150 + 0.462i)7-s + (−2.88 − 8.89i)8-s + (−0.809 + 0.587i)9-s + (3.06 + 1.26i)11-s + 5.43·12-s + (2.92 − 2.12i)13-s + (0.409 + 1.26i)14-s + (−11.8 − 8.59i)16-s + (−2.94 − 2.14i)17-s + (−0.842 + 2.59i)18-s + (−0.504 − 1.55i)19-s + ⋯
L(s)  = 1  + (1.55 − 1.13i)2-s + (0.178 + 0.549i)3-s + (0.839 − 2.58i)4-s + (0.900 + 0.654i)6-s + (−0.0568 + 0.174i)7-s + (−1.02 − 3.14i)8-s + (−0.269 + 0.195i)9-s + (0.924 + 0.381i)11-s + 1.56·12-s + (0.810 − 0.588i)13-s + (0.109 + 0.336i)14-s + (−2.95 − 2.14i)16-s + (−0.714 − 0.519i)17-s + (−0.198 + 0.611i)18-s + (−0.115 − 0.355i)19-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=((−0.100+0.994i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s+1/2) L(s)=((−0.100+0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: −0.100+0.994i-0.100 + 0.994i
Analytic conductor: 6.587656.58765
Root analytic conductor: 2.566642.56664
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ825(751,⋅)\chi_{825} (751, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 825, ( :1/2), −0.100+0.994i)(2,\ 825,\ (\ :1/2),\ -0.100 + 0.994i)

Particular Values

L(1)L(1) ≈\approx 2.62885−2.90906i2.62885 - 2.90906i
L(12)L(\frac12) ≈\approx 2.62885−2.90906i2.62885 - 2.90906i
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+(−3.06−1.26i)T 1 + (-3.06 - 1.26i)T
good2 1+(−2.20+1.60i)T+(0.618−1.90i)T2 1 + (-2.20 + 1.60i)T + (0.618 - 1.90i)T^{2}
7 1+(0.150−0.462i)T+(−5.66−4.11i)T2 1 + (0.150 - 0.462i)T + (-5.66 - 4.11i)T^{2}
13 1+(−2.92+2.12i)T+(4.01−12.3i)T2 1 + (-2.92 + 2.12i)T + (4.01 - 12.3i)T^{2}
17 1+(2.94+2.14i)T+(5.25+16.1i)T2 1 + (2.94 + 2.14i)T + (5.25 + 16.1i)T^{2}
19 1+(0.504+1.55i)T+(−15.3+11.1i)T2 1 + (0.504 + 1.55i)T + (-15.3 + 11.1i)T^{2}
23 1−3.54T+23T2 1 - 3.54T + 23T^{2}
29 1+(−0.326+1.00i)T+(−23.4−17.0i)T2 1 + (-0.326 + 1.00i)T + (-23.4 - 17.0i)T^{2}
31 1+(4.93−3.58i)T+(9.57−29.4i)T2 1 + (4.93 - 3.58i)T + (9.57 - 29.4i)T^{2}
37 1+(2.65−8.17i)T+(−29.9−21.7i)T2 1 + (2.65 - 8.17i)T + (-29.9 - 21.7i)T^{2}
41 1+(−3.39−10.4i)T+(−33.1+24.0i)T2 1 + (-3.39 - 10.4i)T + (-33.1 + 24.0i)T^{2}
43 1+7.41T+43T2 1 + 7.41T + 43T^{2}
47 1+(1.61+4.95i)T+(−38.0+27.6i)T2 1 + (1.61 + 4.95i)T + (-38.0 + 27.6i)T^{2}
53 1+(1.20−0.875i)T+(16.3−50.4i)T2 1 + (1.20 - 0.875i)T + (16.3 - 50.4i)T^{2}
59 1+(2.46−7.58i)T+(−47.7−34.6i)T2 1 + (2.46 - 7.58i)T + (-47.7 - 34.6i)T^{2}
61 1+(−8.78−6.38i)T+(18.8+58.0i)T2 1 + (-8.78 - 6.38i)T + (18.8 + 58.0i)T^{2}
67 1−0.432T+67T2 1 - 0.432T + 67T^{2}
71 1+(4.86+3.53i)T+(21.9+67.5i)T2 1 + (4.86 + 3.53i)T + (21.9 + 67.5i)T^{2}
73 1+(−0.359+1.10i)T+(−59.0−42.9i)T2 1 + (-0.359 + 1.10i)T + (-59.0 - 42.9i)T^{2}
79 1+(−3.57+2.60i)T+(24.4−75.1i)T2 1 + (-3.57 + 2.60i)T + (24.4 - 75.1i)T^{2}
83 1+(−2.57−1.86i)T+(25.6+78.9i)T2 1 + (-2.57 - 1.86i)T + (25.6 + 78.9i)T^{2}
89 1−1.38T+89T2 1 - 1.38T + 89T^{2}
97 1+(1.13−0.822i)T+(29.9−92.2i)T2 1 + (1.13 - 0.822i)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

−10.30066172168846589229082992854, −9.490627864998833629862411417318, −8.689736847133377902404262204743, −6.95443826901507375215120931241, −6.17105838881627345728777454064, −5.15178536324223104316330929477, −4.45096256549731660216401028039, −3.52587235889781025805282356891, −2.72436009367773811788371386498, −1.37098665775371471921761306976, 2.05710247869621753091971931670, 3.57046656757267854981545944903, 4.02258712109308850938013917379, 5.30456819679920883084413702343, 6.19507552133500140926860685554, 6.74745170596026342260478291099, 7.48871528209337333996828013425, 8.525598144341531885677626004844, 9.059181842571559348521869068435, 10.95360364675247013597436403690

Graph of the ZZ-function along the critical line