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Properties

Label 2-334-1.1-c1-0-6
Degree 22
Conductor 334334
Sign 11
Analytic cond. 2.667002.66700
Root an. cond. 1.633091.63309
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 + 1.17·3-s + 4-s − 5-s + 1.17·6-s + 7-s + 8-s − 1.61·9-s − 10-s + 4.25·11-s + 1.17·12-s + 3.43·13-s + 14-s − 1.17·15-s + 16-s + 4.61·17-s − 1.61·18-s − 5.79·19-s − 20-s + 1.17·21-s + 4.25·22-s − 8.65·23-s + 1.17·24-s − 4·25-s + 3.43·26-s − 5.43·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.680·3-s + 0.5·4-s − 0.447·5-s + 0.480·6-s + 0.377·7-s + 0.353·8-s − 0.537·9-s − 0.316·10-s + 1.28·11-s + 0.340·12-s + 0.952·13-s + 0.267·14-s − 0.304·15-s + 0.250·16-s + 1.11·17-s − 0.379·18-s − 1.32·19-s − 0.223·20-s + 0.257·21-s + 0.907·22-s − 1.80·23-s + 0.240·24-s − 0.800·25-s + 0.673·26-s − 1.04·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(334s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(334s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 334334    =    2⋅1672 \cdot 167
Sign: 11
Analytic conductor: 2.667002.66700
Root analytic conductor: 1.633091.63309
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 334, ( :1/2), 1)(2,\ 334,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.3524300202.352430020
L(12)L(\frac12) ≈\approx 2.3524300202.352430020
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
167 1+T 1 + T
good3 1−1.17T+3T2 1 - 1.17T + 3T^{2}
5 1+T+5T2 1 + T + 5T^{2}
7 1−T+7T2 1 - T + 7T^{2}
11 1−4.25T+11T2 1 - 4.25T + 11T^{2}
13 1−3.43T+13T2 1 - 3.43T + 13T^{2}
17 1−4.61T+17T2 1 - 4.61T + 17T^{2}
19 1+5.79T+19T2 1 + 5.79T + 19T^{2}
23 1+8.65T+23T2 1 + 8.65T + 23T^{2}
29 1+1.43T+29T2 1 + 1.43T + 29T^{2}
31 1−3.86T+31T2 1 - 3.86T + 31T^{2}
37 1+3.25T+37T2 1 + 3.25T + 37T^{2}
41 1+3.43T+41T2 1 + 3.43T + 41T^{2}
43 1+10.4T+43T2 1 + 10.4T + 43T^{2}
47 1−12.0T+47T2 1 - 12.0T + 47T^{2}
53 1−7.82T+53T2 1 - 7.82T + 53T^{2}
59 1+4.22T+59T2 1 + 4.22T + 59T^{2}
61 1+13.2T+61T2 1 + 13.2T + 61T^{2}
67 1−15.1T+67T2 1 - 15.1T + 67T^{2}
71 1−8.86T+71T2 1 - 8.86T + 71T^{2}
73 1−2.40T+73T2 1 - 2.40T + 73T^{2}
79 1−14.6T+79T2 1 - 14.6T + 79T^{2}
83 1+1.46T+83T2 1 + 1.46T + 83T^{2}
89 1+7.14T+89T2 1 + 7.14T + 89T^{2}
97 1+17.3T+97T2 1 + 17.3T + 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

−11.80064522074982062923253944016, −10.86947952739194722489204994603, −9.686233204477995910583280532148, −8.459843000246331077527942259582, −7.993865581387037858329963090199, −6.56871535579198105366446357127, −5.70510342973306760500478597807, −4.11898576121633435982316952097, −3.52050833451732964621868825321, −1.89290360434619087347588841501, 1.89290360434619087347588841501, 3.52050833451732964621868825321, 4.11898576121633435982316952097, 5.70510342973306760500478597807, 6.56871535579198105366446357127, 7.993865581387037858329963090199, 8.459843000246331077527942259582, 9.686233204477995910583280532148, 10.86947952739194722489204994603, 11.80064522074982062923253944016

Graph of the ZZ-function along the critical line