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Properties

Label 2-975-65.29-c1-0-32
Degree 22
Conductor 975975
Sign 0.450+0.892i0.450 + 0.892i
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  + (2.21 − 1.28i)2-s + (−0.866 + 0.5i)3-s + (2.28 − 3.95i)4-s + (−1.28 + 2.21i)6-s + (3.08 + 1.78i)7-s − 6.56i·8-s + (0.499 − 0.866i)9-s + (1 + 1.73i)11-s + 4.56i·12-s + (3.57 − 0.5i)13-s + 9.12·14-s + (−3.84 − 6.65i)16-s + (−2.21 − 1.28i)17-s − 2.56i·18-s + (−0.561 + 0.972i)19-s + ⋯
L(s)  = 1  + (1.56 − 0.905i)2-s + (−0.499 + 0.288i)3-s + (1.14 − 1.97i)4-s + (−0.522 + 0.905i)6-s + (1.16 + 0.673i)7-s − 2.31i·8-s + (0.166 − 0.288i)9-s + (0.301 + 0.522i)11-s + 1.31i·12-s + (0.990 − 0.138i)13-s + 2.43·14-s + (−0.960 − 1.66i)16-s + (−0.538 − 0.310i)17-s − 0.603i·18-s + (−0.128 + 0.223i)19-s + ⋯

Functional equation

Λ(s)=(975s/2 ΓC(s) L(s)=((0.450+0.892i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(975s/2 ΓC(s+1/2) L(s)=((0.450+0.892i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 975975    =    3⋅52⋅133 \cdot 5^{2} \cdot 13
Sign: 0.450+0.892i0.450 + 0.892i
Analytic conductor: 7.785417.78541
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ975(874,⋅)\chi_{975} (874, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 975, ( :1/2), 0.450+0.892i)(2,\ 975,\ (\ :1/2),\ 0.450 + 0.892i)

Particular Values

L(1)L(1) ≈\approx 3.23921−1.99380i3.23921 - 1.99380i
L(12)L(\frac12) ≈\approx 3.23921−1.99380i3.23921 - 1.99380i
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+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
5 1 1
13 1+(−3.57+0.5i)T 1 + (-3.57 + 0.5i)T
good2 1+(−2.21+1.28i)T+(1−1.73i)T2 1 + (-2.21 + 1.28i)T + (1 - 1.73i)T^{2}
7 1+(−3.08−1.78i)T+(3.5+6.06i)T2 1 + (-3.08 - 1.78i)T + (3.5 + 6.06i)T^{2}
11 1+(−1−1.73i)T+(−5.5+9.52i)T2 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2}
17 1+(2.21+1.28i)T+(8.5+14.7i)T2 1 + (2.21 + 1.28i)T + (8.5 + 14.7i)T^{2}
19 1+(0.561−0.972i)T+(−9.5−16.4i)T2 1 + (0.561 - 0.972i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.73−i)T+(11.5−19.9i)T2 1 + (1.73 - i)T + (11.5 - 19.9i)T^{2}
29 1+(2.84+4.92i)T+(−14.5+25.1i)T2 1 + (2.84 + 4.92i)T + (-14.5 + 25.1i)T^{2}
31 1+1.56T+31T2 1 + 1.56T + 31T^{2}
37 1+(−2.97+1.71i)T+(18.5−32.0i)T2 1 + (-2.97 + 1.71i)T + (18.5 - 32.0i)T^{2}
41 1+(1.28+2.21i)T+(−20.5+35.5i)T2 1 + (1.28 + 2.21i)T + (-20.5 + 35.5i)T^{2}
43 1+(−0.379−0.219i)T+(21.5+37.2i)T2 1 + (-0.379 - 0.219i)T + (21.5 + 37.2i)T^{2}
47 1+8.24iT−47T2 1 + 8.24iT - 47T^{2}
53 1+11.6iT−53T2 1 + 11.6iT - 53T^{2}
59 1+(5.56−9.63i)T+(−29.5−51.0i)T2 1 + (5.56 - 9.63i)T + (-29.5 - 51.0i)T^{2}
61 1+(6.06−10.4i)T+(−30.5−52.8i)T2 1 + (6.06 - 10.4i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.379+0.219i)T+(33.5−58.0i)T2 1 + (-0.379 + 0.219i)T + (33.5 - 58.0i)T^{2}
71 1+(7−12.1i)T+(−35.5−61.4i)T2 1 + (7 - 12.1i)T + (-35.5 - 61.4i)T^{2}
73 1−1.87iT−73T2 1 - 1.87iT - 73T^{2}
79 1+9.56T+79T2 1 + 9.56T + 79T^{2}
83 1−9.12iT−83T2 1 - 9.12iT - 83T^{2}
89 1+(−6.56−11.3i)T+(−44.5+77.0i)T2 1 + (-6.56 - 11.3i)T + (-44.5 + 77.0i)T^{2}
97 1+(−3.84−2.21i)T+(48.5+84.0i)T2 1 + (-3.84 - 2.21i)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

−10.33406379978787693962128654566, −9.305147586573910571060499595304, −8.243162938039922743841372916094, −6.91697214075337980506260824994, −5.87587823253042500150131451295, −5.38143042501547737118817396027, −4.43171502086696322807791983433, −3.82533939277645362843121951178, −2.42242174595223865497055331108, −1.48692520106216810671316459752, 1.60360467727692540339279648819, 3.27673836410607749528869622382, 4.30426859820601916826247569271, 4.84052266981001478327102447705, 5.95152448154802315811676070114, 6.41662221913434678374831985197, 7.43131096360612999864611153862, 8.010350929940172427547605295191, 8.960256857157523890267652242431, 10.74960932783229664605515057565

Graph of the ZZ-function along the critical line