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Properties

Label 2-975-975.191-c0-0-1
Degree 22
Conductor 975975
Sign −0.943+0.332i-0.943 + 0.332i
Analytic cond. 0.4865880.486588
Root an. cond. 0.6975580.697558
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.251 − 0.564i)2-s + (0.207 − 0.978i)3-s + (0.413 − 0.459i)4-s + (−0.587 + 0.809i)5-s + (−0.604 + 0.128i)6-s + (−0.5 − 0.866i)7-s + (−0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + (0.604 + 0.128i)10-s + (0.251 + 0.564i)11-s + (−0.363 − 0.500i)12-s + (0.809 − 0.587i)13-s + (−0.363 + 0.5i)14-s + (0.669 + 0.743i)15-s + (−0.207 − 0.978i)17-s + 0.618i·18-s + ⋯
L(s)  = 1  + (−0.251 − 0.564i)2-s + (0.207 − 0.978i)3-s + (0.413 − 0.459i)4-s + (−0.587 + 0.809i)5-s + (−0.604 + 0.128i)6-s + (−0.5 − 0.866i)7-s + (−0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + (0.604 + 0.128i)10-s + (0.251 + 0.564i)11-s + (−0.363 − 0.500i)12-s + (0.809 − 0.587i)13-s + (−0.363 + 0.5i)14-s + (0.669 + 0.743i)15-s + (−0.207 − 0.978i)17-s + 0.618i·18-s + ⋯

Functional equation

Λ(s)=(975s/2 ΓC(s) L(s)=((−0.943+0.332i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(975s/2 ΓC(s) L(s)=((−0.943+0.332i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 975975    =    3⋅52⋅133 \cdot 5^{2} \cdot 13
Sign: −0.943+0.332i-0.943 + 0.332i
Analytic conductor: 0.4865880.486588
Root analytic conductor: 0.6975580.697558
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ975(191,⋅)\chi_{975} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 975, ( :0), −0.943+0.332i)(2,\ 975,\ (\ :0),\ -0.943 + 0.332i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.78449101570.7844910157
L(12)L(\frac12) ≈\approx 0.78449101570.7844910157
L(1)L(1) 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+(−0.207+0.978i)T 1 + (-0.207 + 0.978i)T
5 1+(0.587−0.809i)T 1 + (0.587 - 0.809i)T
13 1+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
good2 1+(0.251+0.564i)T+(−0.669+0.743i)T2 1 + (0.251 + 0.564i)T + (-0.669 + 0.743i)T^{2}
7 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
11 1+(−0.251−0.564i)T+(−0.669+0.743i)T2 1 + (-0.251 - 0.564i)T + (-0.669 + 0.743i)T^{2}
17 1+(0.207+0.978i)T+(−0.913+0.406i)T2 1 + (0.207 + 0.978i)T + (-0.913 + 0.406i)T^{2}
19 1+(0.978−0.207i)T+(0.913−0.406i)T2 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2}
23 1+(0.406+0.913i)T+(−0.669+0.743i)T2 1 + (0.406 + 0.913i)T + (-0.669 + 0.743i)T^{2}
29 1+(−0.913−0.406i)T2 1 + (-0.913 - 0.406i)T^{2}
31 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
37 1+(−0.104−0.994i)T+(−0.978+0.207i)T2 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2}
41 1+(−0.994+0.104i)T+(0.978−0.207i)T2 1 + (-0.994 + 0.104i)T + (0.978 - 0.207i)T^{2}
43 1+(−0.809−1.40i)T+(−0.5+0.866i)T2 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2}
47 1+(−1.53+0.5i)T+(0.809−0.587i)T2 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2}
53 1+(−0.587+0.190i)T+(0.809−0.587i)T2 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2}
59 1+(−0.406+0.913i)T+(−0.669−0.743i)T2 1 + (-0.406 + 0.913i)T + (-0.669 - 0.743i)T^{2}
61 1+(0.104−0.994i)T+(−0.978−0.207i)T2 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2}
67 1+(1.08+1.20i)T+(−0.104+0.994i)T2 1 + (1.08 + 1.20i)T + (-0.104 + 0.994i)T^{2}
71 1+(0.104+0.994i)T2 1 + (0.104 + 0.994i)T^{2}
73 1+(−0.809−0.587i)T+(0.309+0.951i)T2 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2}
79 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
83 1+(0.951+0.309i)T+(0.809+0.587i)T2 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2}
89 1+(−0.406−0.913i)T+(−0.669+0.743i)T2 1 + (-0.406 - 0.913i)T + (-0.669 + 0.743i)T^{2}
97 1+(−1.08+1.20i)T+(−0.104−0.994i)T2 1 + (-1.08 + 1.20i)T + (-0.104 - 0.994i)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

−10.09907407730014913519378227822, −9.069074291870934160609035036857, −8.050144267686844568044041141087, −7.17671030889853652791174909802, −6.60990172781766522506778373710, −5.95387989751729011334127145671, −4.17459676000513342035857318275, −3.10819767468947019993254109937, −2.28420086772296671246306701482, −0.77279085316847595321165205503, 2.34123269494058242456568465982, 3.64624413130197759045229735645, 4.16697920058513164205378931576, 5.74176194468989210122297078265, 6.02870737380203440423285315919, 7.40884524532634466958871710220, 8.374354746870385045982725337037, 8.919643646920828908239974939347, 9.189715110245934912105592950732, 10.65733265960111368930100550141

Graph of the ZZ-function along the critical line