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Properties

Label 2-975-39.5-c1-0-78
Degree 22
Conductor 975975
Sign −0.614−0.789i-0.614 - 0.789i
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  + (−0.707 − 0.707i)2-s + (1 − 1.41i)3-s − 0.999i·4-s + (−1.70 + 0.292i)6-s + (−1 − i)7-s + (−2.12 + 2.12i)8-s + (−1.00 − 2.82i)9-s + (−2.82 + 2.82i)11-s + (−1.41 − 0.999i)12-s + (2 − 3i)13-s + 1.41i·14-s + 1.00·16-s + (−1.29 + 2.70i)18-s + (1 − i)19-s + (−2.41 + 0.414i)21-s + 4.00·22-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.577 − 0.816i)3-s − 0.499i·4-s + (−0.696 + 0.119i)6-s + (−0.377 − 0.377i)7-s + (−0.750 + 0.750i)8-s + (−0.333 − 0.942i)9-s + (−0.852 + 0.852i)11-s + (−0.408 − 0.288i)12-s + (0.554 − 0.832i)13-s + 0.377i·14-s + 0.250·16-s + (−0.304 + 0.638i)18-s + (0.229 − 0.229i)19-s + (−0.526 + 0.0903i)21-s + 0.852·22-s + ⋯

Functional equation

Λ(s)=(975s/2 ΓC(s) L(s)=((−0.614−0.789i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(975s/2 ΓC(s+1/2) L(s)=((−0.614−0.789i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 975975    =    3⋅52⋅133 \cdot 5^{2} \cdot 13
Sign: −0.614−0.789i-0.614 - 0.789i
Analytic conductor: 7.785417.78541
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ975(551,⋅)\chi_{975} (551, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 975, ( :1/2), −0.614−0.789i)(2,\ 975,\ (\ :1/2),\ -0.614 - 0.789i)

Particular Values

L(1)L(1) ≈\approx 0.271828+0.555983i0.271828 + 0.555983i
L(12)L(\frac12) ≈\approx 0.271828+0.555983i0.271828 + 0.555983i
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+1.41i)T 1 + (-1 + 1.41i)T
5 1 1
13 1+(−2+3i)T 1 + (-2 + 3i)T
good2 1+(0.707+0.707i)T+2iT2 1 + (0.707 + 0.707i)T + 2iT^{2}
7 1+(1+i)T+7iT2 1 + (1 + i)T + 7iT^{2}
11 1+(2.82−2.82i)T−11iT2 1 + (2.82 - 2.82i)T - 11iT^{2}
17 1+17T2 1 + 17T^{2}
19 1+(−1+i)T−19iT2 1 + (-1 + i)T - 19iT^{2}
23 1+8.48T+23T2 1 + 8.48T + 23T^{2}
29 1+2.82iT−29T2 1 + 2.82iT - 29T^{2}
31 1+(5−5i)T−31iT2 1 + (5 - 5i)T - 31iT^{2}
37 1+(1+i)T+37iT2 1 + (1 + i)T + 37iT^{2}
41 1+(1.41+1.41i)T+41iT2 1 + (1.41 + 1.41i)T + 41iT^{2}
43 1−6iT−43T2 1 - 6iT - 43T^{2}
47 1+(−2.82+2.82i)T−47iT2 1 + (-2.82 + 2.82i)T - 47iT^{2}
53 1+5.65iT−53T2 1 + 5.65iT - 53T^{2}
59 1+(2.82−2.82i)T−59iT2 1 + (2.82 - 2.82i)T - 59iT^{2}
61 1−8T+61T2 1 - 8T + 61T^{2}
67 1+(−5+5i)T−67iT2 1 + (-5 + 5i)T - 67iT^{2}
71 1+(−2.82−2.82i)T+71iT2 1 + (-2.82 - 2.82i)T + 71iT^{2}
73 1+(1+i)T+73iT2 1 + (1 + i)T + 73iT^{2}
79 1+10T+79T2 1 + 10T + 79T^{2}
83 1+(−5.65−5.65i)T+83iT2 1 + (-5.65 - 5.65i)T + 83iT^{2}
89 1+(−9.89+9.89i)T−89iT2 1 + (-9.89 + 9.89i)T - 89iT^{2}
97 1+(7−7i)T−97iT2 1 + (7 - 7i)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

−9.673451268081745190128082809495, −8.599061143618181068014766861767, −7.961762332918055513978201182778, −7.07016635259605825682806147076, −6.11303450444885643996722076195, −5.26020901751273711743845626171, −3.73149944900423468378206603820, −2.61279009361964217256241870082, −1.69219593981253661912035641300, −0.29505375134976694326419730739, 2.35446875000867515281987056864, 3.42587723280838639040613155157, 4.10649225898848003879516446271, 5.51681358829249898529448746781, 6.28587054128424301410234678989, 7.51766203852099923779310812915, 8.168509582588643533159577507918, 8.846028324195117207948416607887, 9.456411782061042449518536235933, 10.28685822042339847621450064783

Graph of the ZZ-function along the critical line