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Properties

Label 2-975-195.38-c1-0-2
Degree 22
Conductor 975975
Sign −0.960−0.277i-0.960 - 0.277i
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  + (−1.25 + 1.25i)2-s + (1.06 − 1.36i)3-s − 1.15i·4-s + (0.381 + 3.05i)6-s + (−1.60 − 1.60i)7-s + (−1.05 − 1.05i)8-s + (−0.738 − 2.90i)9-s − 2.48·11-s + (−1.58 − 1.23i)12-s + (−0.624 + 3.55i)13-s + 4.02·14-s + 4.97·16-s + (1.27 + 1.27i)17-s + (4.58 + 2.72i)18-s + 1.93·19-s + ⋯
L(s)  = 1  + (−0.888 + 0.888i)2-s + (0.613 − 0.789i)3-s − 0.579i·4-s + (0.155 + 1.24i)6-s + (−0.605 − 0.605i)7-s + (−0.373 − 0.373i)8-s + (−0.246 − 0.969i)9-s − 0.748·11-s + (−0.457 − 0.355i)12-s + (−0.173 + 0.984i)13-s + 1.07·14-s + 1.24·16-s + (0.309 + 0.309i)17-s + (1.08 + 0.642i)18-s + 0.443·19-s + ⋯

Functional equation

Λ(s)=(975s/2 ΓC(s) L(s)=((−0.960−0.277i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(975s/2 ΓC(s+1/2) L(s)=((−0.960−0.277i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 975975    =    3⋅52⋅133 \cdot 5^{2} \cdot 13
Sign: −0.960−0.277i-0.960 - 0.277i
Analytic conductor: 7.785417.78541
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ975(818,⋅)\chi_{975} (818, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 975, ( :1/2), −0.960−0.277i)(2,\ 975,\ (\ :1/2),\ -0.960 - 0.277i)

Particular Values

L(1)L(1) ≈\approx 0.0413133+0.291456i0.0413133 + 0.291456i
L(12)L(\frac12) ≈\approx 0.0413133+0.291456i0.0413133 + 0.291456i
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+(−1.06+1.36i)T 1 + (-1.06 + 1.36i)T
5 1 1
13 1+(0.624−3.55i)T 1 + (0.624 - 3.55i)T
good2 1+(1.25−1.25i)T−2iT2 1 + (1.25 - 1.25i)T - 2iT^{2}
7 1+(1.60+1.60i)T+7iT2 1 + (1.60 + 1.60i)T + 7iT^{2}
11 1+2.48T+11T2 1 + 2.48T + 11T^{2}
17 1+(−1.27−1.27i)T+17iT2 1 + (-1.27 - 1.27i)T + 17iT^{2}
19 1−1.93T+19T2 1 - 1.93T + 19T^{2}
23 1+(5.56−5.56i)T−23iT2 1 + (5.56 - 5.56i)T - 23iT^{2}
29 1+9.66T+29T2 1 + 9.66T + 29T^{2}
31 1−6.61iT−31T2 1 - 6.61iT - 31T^{2}
37 1+(−3.53−3.53i)T+37iT2 1 + (-3.53 - 3.53i)T + 37iT^{2}
41 1−7.35T+41T2 1 - 7.35T + 41T^{2}
43 1+(−3.46−3.46i)T+43iT2 1 + (-3.46 - 3.46i)T + 43iT^{2}
47 1+(4.55−4.55i)T−47iT2 1 + (4.55 - 4.55i)T - 47iT^{2}
53 1+(3.97−3.97i)T−53iT2 1 + (3.97 - 3.97i)T - 53iT^{2}
59 1−2.79iT−59T2 1 - 2.79iT - 59T^{2}
61 1+2.54T+61T2 1 + 2.54T + 61T^{2}
67 1+(1.34+1.34i)T+67iT2 1 + (1.34 + 1.34i)T + 67iT^{2}
71 1+6.58T+71T2 1 + 6.58T + 71T^{2}
73 1+(−2.82+2.82i)T−73iT2 1 + (-2.82 + 2.82i)T - 73iT^{2}
79 1−6.48iT−79T2 1 - 6.48iT - 79T^{2}
83 1+(−3.17−3.17i)T+83iT2 1 + (-3.17 - 3.17i)T + 83iT^{2}
89 1+7.47iT−89T2 1 + 7.47iT - 89T^{2}
97 1+(9.27+9.27i)T+97iT2 1 + (9.27 + 9.27i)T + 97iT^{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

−9.755267417142640885638714560506, −9.528353430820723325273536269855, −8.552858780310934844310825613597, −7.60820195491472939458927146581, −7.41994941436599598412400973541, −6.44948523757964435867371978689, −5.71333891353893561985636157964, −3.97590179327058235844161674614, −3.03938440693348914067603044916, −1.49444729401967876410334665141, 0.16389591970516038518976631067, 2.24602124782070832764725456518, 2.82318370510624019043212219860, 3.87225011594347989953313853466, 5.31548698811857014486214918766, 5.91364971811874196700541047310, 7.73524278470630851516203525694, 8.107287523078234370733049046110, 9.285752601024123225362558836624, 9.502127132796374542156665677963

Graph of the ZZ-function along the critical line