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Properties

Label 2-975-1.1-c1-0-23
Degree 22
Conductor 975975
Sign 11
Analytic cond. 7.785417.78541
Root an. cond. 2.790232.79023
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.53·2-s + 3-s + 0.369·4-s + 1.53·6-s + 3.87·7-s − 2.51·8-s + 9-s + 1.24·11-s + 0.369·12-s − 13-s + 5.97·14-s − 4.60·16-s + 0.659·17-s + 1.53·18-s + 6.97·19-s + 3.87·21-s + 1.92·22-s + 1.55·23-s − 2.51·24-s − 1.53·26-s + 27-s + 1.43·28-s − 3·29-s + 5.43·31-s − 2.06·32-s + 1.24·33-s + 1.01·34-s + ⋯
L(s)  = 1  + 1.08·2-s + 0.577·3-s + 0.184·4-s + 0.628·6-s + 1.46·7-s − 0.887·8-s + 0.333·9-s + 0.376·11-s + 0.106·12-s − 0.277·13-s + 1.59·14-s − 1.15·16-s + 0.160·17-s + 0.362·18-s + 1.59·19-s + 0.846·21-s + 0.409·22-s + 0.323·23-s − 0.512·24-s − 0.301·26-s + 0.192·27-s + 0.270·28-s − 0.557·29-s + 0.975·31-s − 0.364·32-s + 0.217·33-s + 0.174·34-s + ⋯

Functional equation

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

Invariants

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

Particular Values

L(1)L(1) ≈\approx 3.5246611123.524661112
L(12)L(\frac12) ≈\approx 3.5246611123.524661112
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−T 1 - T
5 1 1
13 1+T 1 + T
good2 1−1.53T+2T2 1 - 1.53T + 2T^{2}
7 1−3.87T+7T2 1 - 3.87T + 7T^{2}
11 1−1.24T+11T2 1 - 1.24T + 11T^{2}
17 1−0.659T+17T2 1 - 0.659T + 17T^{2}
19 1−6.97T+19T2 1 - 6.97T + 19T^{2}
23 1−1.55T+23T2 1 - 1.55T + 23T^{2}
29 1+3T+29T2 1 + 3T + 29T^{2}
31 1−5.43T+31T2 1 - 5.43T + 31T^{2}
37 1−2.29T+37T2 1 - 2.29T + 37T^{2}
41 1+10.2T+41T2 1 + 10.2T + 41T^{2}
43 1+8.20T+43T2 1 + 8.20T + 43T^{2}
47 1−1.53T+47T2 1 - 1.53T + 47T^{2}
53 1+1.44T+53T2 1 + 1.44T + 53T^{2}
59 1+12.0T+59T2 1 + 12.0T + 59T^{2}
61 1+12.1T+61T2 1 + 12.1T + 61T^{2}
67 1−9.92T+67T2 1 - 9.92T + 67T^{2}
71 1−11.6T+71T2 1 - 11.6T + 71T^{2}
73 1+6.20T+73T2 1 + 6.20T + 73T^{2}
79 1+0.474T+79T2 1 + 0.474T + 79T^{2}
83 1−13.4T+83T2 1 - 13.4T + 83T^{2}
89 1+15.9T+89T2 1 + 15.9T + 89T^{2}
97 1−11.6T+97T2 1 - 11.6T + 97T^{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

−9.901432876215889903039653369273, −9.136070051437953248892360602652, −8.261738277124373502898319837219, −7.54925726713571175162821961570, −6.47149338105626105249361399572, −5.23241160401686095331487935997, −4.83948331932026341695916922314, −3.78735734705806084789356953927, −2.86266284484257058009480957942, −1.49408254883864977709379270130, 1.49408254883864977709379270130, 2.86266284484257058009480957942, 3.78735734705806084789356953927, 4.83948331932026341695916922314, 5.23241160401686095331487935997, 6.47149338105626105249361399572, 7.54925726713571175162821961570, 8.261738277124373502898319837219, 9.136070051437953248892360602652, 9.901432876215889903039653369273

Graph of the ZZ-function along the critical line