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Properties

Label 2-975-13.3-c1-0-42
Degree 22
Conductor 975975
Sign −0.767−0.641i-0.767 - 0.641i
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  + (−0.115 − 0.200i)2-s + (−0.5 − 0.866i)3-s + (0.973 − 1.68i)4-s + (−0.115 + 0.200i)6-s + (−0.580 + 1.00i)7-s − 0.914·8-s + (−0.499 + 0.866i)9-s + (−1.76 − 3.05i)11-s − 1.94·12-s + (−3.59 + 0.297i)13-s + 0.269·14-s + (−1.84 − 3.18i)16-s + (−3.08 + 5.34i)17-s + 0.231·18-s + (−3.63 + 6.29i)19-s + ⋯
L(s)  = 1  + (−0.0819 − 0.141i)2-s + (−0.288 − 0.499i)3-s + (0.486 − 0.842i)4-s + (−0.0472 + 0.0819i)6-s + (−0.219 + 0.380i)7-s − 0.323·8-s + (−0.166 + 0.288i)9-s + (−0.531 − 0.920i)11-s − 0.561·12-s + (−0.996 + 0.0824i)13-s + 0.0719·14-s + (−0.460 − 0.796i)16-s + (−0.748 + 1.29i)17-s + 0.0546·18-s + (−0.834 + 1.44i)19-s + ⋯

Functional equation

Λ(s)=(975s/2 ΓC(s) L(s)=((−0.767−0.641i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(975s/2 ΓC(s+1/2) L(s)=((−0.767−0.641i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.767 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 975975    =    3⋅52⋅133 \cdot 5^{2} \cdot 13
Sign: −0.767−0.641i-0.767 - 0.641i
Analytic conductor: 7.785417.78541
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ975(601,⋅)\chi_{975} (601, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 975, ( :1/2), −0.767−0.641i)(2,\ 975,\ (\ :1/2),\ -0.767 - 0.641i)

Particular Values

L(1)L(1) ≈\approx 0.0811961+0.223679i0.0811961 + 0.223679i
L(12)L(\frac12) ≈\approx 0.0811961+0.223679i0.0811961 + 0.223679i
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+(3.59−0.297i)T 1 + (3.59 - 0.297i)T
good2 1+(0.115+0.200i)T+(−1+1.73i)T2 1 + (0.115 + 0.200i)T + (-1 + 1.73i)T^{2}
7 1+(0.580−1.00i)T+(−3.5−6.06i)T2 1 + (0.580 - 1.00i)T + (-3.5 - 6.06i)T^{2}
11 1+(1.76+3.05i)T+(−5.5+9.52i)T2 1 + (1.76 + 3.05i)T + (-5.5 + 9.52i)T^{2}
17 1+(3.08−5.34i)T+(−8.5−14.7i)T2 1 + (3.08 - 5.34i)T + (-8.5 - 14.7i)T^{2}
19 1+(3.63−6.29i)T+(−9.5−16.4i)T2 1 + (3.63 - 6.29i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.180+0.313i)T+(−11.5+19.9i)T2 1 + (0.180 + 0.313i)T + (-11.5 + 19.9i)T^{2}
29 1+(4.95+8.59i)T+(−14.5+25.1i)T2 1 + (4.95 + 8.59i)T + (-14.5 + 25.1i)T^{2}
31 1−8.83T+31T2 1 - 8.83T + 31T^{2}
37 1+(1.85+3.21i)T+(−18.5+32.0i)T2 1 + (1.85 + 3.21i)T + (-18.5 + 32.0i)T^{2}
41 1+(3.08+5.34i)T+(−20.5+35.5i)T2 1 + (3.08 + 5.34i)T + (-20.5 + 35.5i)T^{2}
43 1+(−1.59+2.76i)T+(−21.5−37.2i)T2 1 + (-1.59 + 2.76i)T + (-21.5 - 37.2i)T^{2}
47 1+11.7T+47T2 1 + 11.7T + 47T^{2}
53 1−3.46T+53T2 1 - 3.46T + 53T^{2}
59 1+(2.74−4.75i)T+(−29.5−51.0i)T2 1 + (2.74 - 4.75i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.361+0.626i)T+(−30.5−52.8i)T2 1 + (-0.361 + 0.626i)T + (-30.5 - 52.8i)T^{2}
67 1+(−2.56−4.44i)T+(−33.5+58.0i)T2 1 + (-2.56 - 4.44i)T + (-33.5 + 58.0i)T^{2}
71 1+(−0.237+0.411i)T+(−35.5−61.4i)T2 1 + (-0.237 + 0.411i)T + (-35.5 - 61.4i)T^{2}
73 1−3.75T+73T2 1 - 3.75T + 73T^{2}
79 1+4.59T+79T2 1 + 4.59T + 79T^{2}
83 1+8.41T+83T2 1 + 8.41T + 83T^{2}
89 1+(−7.70−13.3i)T+(−44.5+77.0i)T2 1 + (-7.70 - 13.3i)T + (-44.5 + 77.0i)T^{2}
97 1+(−0.662+1.14i)T+(−48.5−84.0i)T2 1 + (-0.662 + 1.14i)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

−9.732627592420544845089197483767, −8.563015356672765163474350984088, −7.889003113867455062050755060551, −6.71483838765953644424125595062, −6.02457648945520579849916325234, −5.49918167453391357181823179994, −4.15133550425370072344721440607, −2.62392310482534199121332457204, −1.79194096687364081388391574887, −0.10436225394837667097878517923, 2.34447772205472033280282092731, 3.16958158876011045177931077731, 4.56862309270662781726831918029, 4.98676881525571720357368501420, 6.69876028223341360490224115767, 6.95154868737803947850484604635, 7.909870522009544868583968942744, 8.926889515071851260295911098726, 9.681042844586958191887890702762, 10.50218156131832270276822903586

Graph of the ZZ-function along the critical line