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Properties

Label 2-334-1.1-c1-0-3
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 − 0.772·3-s + 4-s + 2.62·5-s + 0.772·6-s + 2.62·7-s − 8-s − 2.40·9-s − 2.62·10-s + 0.597·11-s − 0.772·12-s + 0.175·13-s − 2.62·14-s − 2.03·15-s + 16-s + 6.94·17-s + 2.40·18-s − 5.72·19-s + 2.62·20-s − 2.03·21-s − 0.597·22-s + 0.175·23-s + 0.772·24-s + 1.91·25-s − 0.175·26-s + 4.17·27-s + 2.62·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.446·3-s + 0.5·4-s + 1.17·5-s + 0.315·6-s + 0.993·7-s − 0.353·8-s − 0.800·9-s − 0.831·10-s + 0.180·11-s − 0.223·12-s + 0.0486·13-s − 0.702·14-s − 0.524·15-s + 0.250·16-s + 1.68·17-s + 0.566·18-s − 1.31·19-s + 0.588·20-s − 0.443·21-s − 0.127·22-s + 0.0366·23-s + 0.157·24-s + 0.383·25-s − 0.0344·26-s + 0.803·27-s + 0.496·28-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 1.0888867811.088886781
L(12)L(\frac12) ≈\approx 1.0888867811.088886781
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+0.772T+3T2 1 + 0.772T + 3T^{2}
5 1−2.62T+5T2 1 - 2.62T + 5T^{2}
7 1−2.62T+7T2 1 - 2.62T + 7T^{2}
11 1−0.597T+11T2 1 - 0.597T + 11T^{2}
13 1−0.175T+13T2 1 - 0.175T + 13T^{2}
17 1−6.94T+17T2 1 - 6.94T + 17T^{2}
19 1+5.72T+19T2 1 + 5.72T + 19T^{2}
23 1−0.175T+23T2 1 - 0.175T + 23T^{2}
29 1−8.62T+29T2 1 - 8.62T + 29T^{2}
31 1−10.9T+31T2 1 - 10.9T + 31T^{2}
37 1−0.772T+37T2 1 - 0.772T + 37T^{2}
41 1−1.82T+41T2 1 - 1.82T + 41T^{2}
43 1+11.1T+43T2 1 + 11.1T + 43T^{2}
47 1+6.80T+47T2 1 + 6.80T + 47T^{2}
53 1−3.05T+53T2 1 - 3.05T + 53T^{2}
59 1−0.629T+59T2 1 - 0.629T + 59T^{2}
61 1−9.92T+61T2 1 - 9.92T + 61T^{2}
67 1+9.22T+67T2 1 + 9.22T + 67T^{2}
71 1+4T+71T2 1 + 4T + 71T^{2}
73 1+13.2T+73T2 1 + 13.2T + 73T^{2}
79 1+9.43T+79T2 1 + 9.43T + 79T^{2}
83 1+9.40T+83T2 1 + 9.40T + 83T^{2}
89 1−17.1T+89T2 1 - 17.1T + 89T^{2}
97 1+14.4T+97T2 1 + 14.4T + 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

−11.53886503261533368655411124214, −10.37336236014094733029761395797, −9.975743970542972682253605327197, −8.615258997960972993392054725636, −8.109069453549852273810996821938, −6.57826486554111031972700048720, −5.84217987258767075933307270228, −4.82306348846688234177878582182, −2.76298440028912646406152942860, −1.36563292974038541742435107004, 1.36563292974038541742435107004, 2.76298440028912646406152942860, 4.82306348846688234177878582182, 5.84217987258767075933307270228, 6.57826486554111031972700048720, 8.109069453549852273810996821938, 8.615258997960972993392054725636, 9.975743970542972682253605327197, 10.37336236014094733029761395797, 11.53886503261533368655411124214

Graph of the ZZ-function along the critical line