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Properties

Label 2-975-13.9-c1-0-12
Degree 22
Conductor 975975
Sign −0.227−0.973i-0.227 - 0.973i
Analytic cond. 7.785417.78541
Root an. cond. 2.790232.79023
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  + (−0.813 + 1.40i)2-s + (−0.5 + 0.866i)3-s + (−0.322 − 0.558i)4-s + (−0.813 − 1.40i)6-s + (−1.54 − 2.67i)7-s − 2.20·8-s + (−0.499 − 0.866i)9-s + (1.60 − 2.78i)11-s + 0.644·12-s + (1.88 − 3.07i)13-s + 5.01·14-s + (2.43 − 4.22i)16-s + (2.01 + 3.48i)17-s + 1.62·18-s + (3.66 + 6.34i)19-s + ⋯
L(s)  = 1  + (−0.574 + 0.995i)2-s + (−0.288 + 0.499i)3-s + (−0.161 − 0.279i)4-s + (−0.331 − 0.574i)6-s + (−0.582 − 1.00i)7-s − 0.779·8-s + (−0.166 − 0.288i)9-s + (0.484 − 0.838i)11-s + 0.186·12-s + (0.522 − 0.852i)13-s + 1.34·14-s + (0.609 − 1.05i)16-s + (0.487 + 0.844i)17-s + 0.383·18-s + (0.840 + 1.45i)19-s + ⋯

Functional equation

Λ(s)=(975s/2 ΓC(s) L(s)=((−0.227−0.973i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(975s/2 ΓC(s+1/2) L(s)=((−0.227−0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 975975    =    3⋅52⋅133 \cdot 5^{2} \cdot 13
Sign: −0.227−0.973i-0.227 - 0.973i
Analytic conductor: 7.785417.78541
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ975(451,⋅)\chi_{975} (451, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 975, ( :1/2), −0.227−0.973i)(2,\ 975,\ (\ :1/2),\ -0.227 - 0.973i)

Particular Values

L(1)L(1) ≈\approx 0.599950+0.755980i0.599950 + 0.755980i
L(12)L(\frac12) ≈\approx 0.599950+0.755980i0.599950 + 0.755980i
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
5 1 1
13 1+(−1.88+3.07i)T 1 + (-1.88 + 3.07i)T
good2 1+(0.813−1.40i)T+(−1−1.73i)T2 1 + (0.813 - 1.40i)T + (-1 - 1.73i)T^{2}
7 1+(1.54+2.67i)T+(−3.5+6.06i)T2 1 + (1.54 + 2.67i)T + (-3.5 + 6.06i)T^{2}
11 1+(−1.60+2.78i)T+(−5.5−9.52i)T2 1 + (-1.60 + 2.78i)T + (-5.5 - 9.52i)T^{2}
17 1+(−2.01−3.48i)T+(−8.5+14.7i)T2 1 + (-2.01 - 3.48i)T + (-8.5 + 14.7i)T^{2}
19 1+(−3.66−6.34i)T+(−9.5+16.4i)T2 1 + (-3.66 - 6.34i)T + (-9.5 + 16.4i)T^{2}
23 1+(3.77−6.54i)T+(−11.5−19.9i)T2 1 + (3.77 - 6.54i)T + (-11.5 - 19.9i)T^{2}
29 1+(2.07−3.58i)T+(−14.5−25.1i)T2 1 + (2.07 - 3.58i)T + (-14.5 - 25.1i)T^{2}
31 1−6.70T+31T2 1 - 6.70T + 31T^{2}
37 1+(2.38−4.13i)T+(−18.5−32.0i)T2 1 + (2.38 - 4.13i)T + (-18.5 - 32.0i)T^{2}
41 1+(−2.01+3.48i)T+(−20.5−35.5i)T2 1 + (-2.01 + 3.48i)T + (-20.5 - 35.5i)T^{2}
43 1+(0.560+0.971i)T+(−21.5+37.2i)T2 1 + (0.560 + 0.971i)T + (-21.5 + 37.2i)T^{2}
47 1−8.97T+47T2 1 - 8.97T + 47T^{2}
53 1−10.0T+53T2 1 - 10.0T + 53T^{2}
59 1+(−2.30−3.98i)T+(−29.5+51.0i)T2 1 + (-2.30 - 3.98i)T + (-29.5 + 51.0i)T^{2}
61 1+(6.51+11.2i)T+(−30.5+52.8i)T2 1 + (6.51 + 11.2i)T + (-30.5 + 52.8i)T^{2}
67 1+(−4.28+7.42i)T+(−33.5−58.0i)T2 1 + (-4.28 + 7.42i)T + (-33.5 - 58.0i)T^{2}
71 1+(−3.60−6.24i)T+(−35.5+61.4i)T2 1 + (-3.60 - 6.24i)T + (-35.5 + 61.4i)T^{2}
73 1−11.9T+73T2 1 - 11.9T + 73T^{2}
79 1+6.12T+79T2 1 + 6.12T + 79T^{2}
83 1−7.24T+83T2 1 - 7.24T + 83T^{2}
89 1+(3.04−5.27i)T+(−44.5−77.0i)T2 1 + (3.04 - 5.27i)T + (-44.5 - 77.0i)T^{2}
97 1+(5.34+9.25i)T+(−48.5+84.0i)T2 1 + (5.34 + 9.25i)T + (-48.5 + 84.0i)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

−10.09174634412610896988791222957, −9.416818917326604801727038806910, −8.343202743784533675298623995249, −7.83531643834330492963951137537, −6.89192345248739009061760223404, −5.95594867245073293918104514336, −5.55470967656795747801387218568, −3.67877709986099843422290326715, −3.48175665404339414113588811360, −0.955117143927324254730013251651, 0.77397962645078541709236786322, 2.18591502973989022224033801254, 2.80593535314417681356980260167, 4.29197538892884274675009956619, 5.57260782418147252302397288458, 6.41538216299602069200083008095, 7.13159232447235800560482403795, 8.451325957506390989511290009735, 9.244552930924283878887337141071, 9.639579424207881931765847356073

Graph of the ZZ-function along the critical line