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Properties

Label 2-678-1.1-c1-0-2
Degree 22
Conductor 678678
Sign 11
Analytic cond. 5.413855.41385
Root an. cond. 2.326762.32676
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  + 2-s − 3-s + 4-s − 3.17·5-s − 6-s − 2.40·7-s + 8-s + 9-s − 3.17·10-s + 5.40·11-s − 12-s + 3.17·13-s − 2.40·14-s + 3.17·15-s + 16-s + 3.94·17-s + 18-s + 3.85·19-s − 3.17·20-s + 2.40·21-s + 5.40·22-s + 5.77·23-s − 24-s + 5.08·25-s + 3.17·26-s − 27-s − 2.40·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.42·5-s − 0.408·6-s − 0.908·7-s + 0.353·8-s + 0.333·9-s − 1.00·10-s + 1.62·11-s − 0.288·12-s + 0.880·13-s − 0.642·14-s + 0.819·15-s + 0.250·16-s + 0.957·17-s + 0.235·18-s + 0.884·19-s − 0.710·20-s + 0.524·21-s + 1.15·22-s + 1.20·23-s − 0.204·24-s + 1.01·25-s + 0.622·26-s − 0.192·27-s − 0.454·28-s + ⋯

Functional equation

Λ(s)=(678s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(678s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 678678    =    2⋅3⋅1132 \cdot 3 \cdot 113
Sign: 11
Analytic conductor: 5.413855.41385
Root analytic conductor: 2.326762.32676
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 678, ( :1/2), 1)(2,\ 678,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5759199221.575919922
L(12)L(\frac12) ≈\approx 1.5759199221.575919922
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)
bad2 1−T 1 - T
3 1+T 1 + T
113 1+T 1 + T
good5 1+3.17T+5T2 1 + 3.17T + 5T^{2}
7 1+2.40T+7T2 1 + 2.40T + 7T^{2}
11 1−5.40T+11T2 1 - 5.40T + 11T^{2}
13 1−3.17T+13T2 1 - 3.17T + 13T^{2}
17 1−3.94T+17T2 1 - 3.94T + 17T^{2}
19 1−3.85T+19T2 1 - 3.85T + 19T^{2}
23 1−5.77T+23T2 1 - 5.77T + 23T^{2}
29 1−1.09T+29T2 1 - 1.09T + 29T^{2}
31 1+7.48T+31T2 1 + 7.48T + 31T^{2}
37 1−3.40T+37T2 1 - 3.40T + 37T^{2}
41 1+4.35T+41T2 1 + 4.35T + 41T^{2}
43 1+5.57T+43T2 1 + 5.57T + 43T^{2}
47 1−3.22T+47T2 1 - 3.22T + 47T^{2}
53 1−6.48T+53T2 1 - 6.48T + 53T^{2}
59 1−3.17T+59T2 1 - 3.17T + 59T^{2}
61 1+11.6T+61T2 1 + 11.6T + 61T^{2}
67 1−8.94T+67T2 1 - 8.94T + 67T^{2}
71 1−9.94T+71T2 1 - 9.94T + 71T^{2}
73 1+4T+73T2 1 + 4T + 73T^{2}
79 1−4.35T+79T2 1 - 4.35T + 79T^{2}
83 1−0.486T+83T2 1 - 0.486T + 83T^{2}
89 1−13.4T+89T2 1 - 13.4T + 89T^{2}
97 1−19.1T+97T2 1 - 19.1T + 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

−10.86926822482350349143050202814, −9.687776569017260204335016985793, −8.774303853161023466515248842862, −7.55245861819499895437604584361, −6.84977765435982612265818462200, −6.06494323702347345190567740065, −4.93116498734303837462335698960, −3.65430670811909668627902852677, −3.49529613509372681109713519604, −1.05919054493459009256996920490, 1.05919054493459009256996920490, 3.49529613509372681109713519604, 3.65430670811909668627902852677, 4.93116498734303837462335698960, 6.06494323702347345190567740065, 6.84977765435982612265818462200, 7.55245861819499895437604584361, 8.774303853161023466515248842862, 9.687776569017260204335016985793, 10.86926822482350349143050202814

Graph of the ZZ-function along the critical line