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Properties

Label 2-1338-1.1-c1-0-8
Degree 22
Conductor 13381338
Sign 11
Analytic cond. 10.683910.6839
Root an. cond. 3.268633.26863
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 + 1.43·5-s − 6-s − 1.87·7-s − 8-s + 9-s − 1.43·10-s + 5.50·11-s + 12-s + 5.81·13-s + 1.87·14-s + 1.43·15-s + 16-s − 3.16·17-s − 18-s − 0.890·19-s + 1.43·20-s − 1.87·21-s − 5.50·22-s + 1.68·23-s − 24-s − 2.94·25-s − 5.81·26-s + 27-s − 1.87·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.641·5-s − 0.408·6-s − 0.707·7-s − 0.353·8-s + 0.333·9-s − 0.453·10-s + 1.66·11-s + 0.288·12-s + 1.61·13-s + 0.500·14-s + 0.370·15-s + 0.250·16-s − 0.766·17-s − 0.235·18-s − 0.204·19-s + 0.320·20-s − 0.408·21-s − 1.17·22-s + 0.351·23-s − 0.204·24-s − 0.588·25-s − 1.13·26-s + 0.192·27-s − 0.353·28-s + ⋯

Functional equation

Λ(s)=(1338s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1338s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13381338    =    2⋅3⋅2232 \cdot 3 \cdot 223
Sign: 11
Analytic conductor: 10.683910.6839
Root analytic conductor: 3.268633.26863
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1338, ( :1/2), 1)(2,\ 1338,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7880672551.788067255
L(12)L(\frac12) ≈\approx 1.7880672551.788067255
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
223 1+T 1 + T
good5 1−1.43T+5T2 1 - 1.43T + 5T^{2}
7 1+1.87T+7T2 1 + 1.87T + 7T^{2}
11 1−5.50T+11T2 1 - 5.50T + 11T^{2}
13 1−5.81T+13T2 1 - 5.81T + 13T^{2}
17 1+3.16T+17T2 1 + 3.16T + 17T^{2}
19 1+0.890T+19T2 1 + 0.890T + 19T^{2}
23 1−1.68T+23T2 1 - 1.68T + 23T^{2}
29 1+6.77T+29T2 1 + 6.77T + 29T^{2}
31 1−2.18T+31T2 1 - 2.18T + 31T^{2}
37 1−9.53T+37T2 1 - 9.53T + 37T^{2}
41 1−5.50T+41T2 1 - 5.50T + 41T^{2}
43 1+9.08T+43T2 1 + 9.08T + 43T^{2}
47 1−4.43T+47T2 1 - 4.43T + 47T^{2}
53 1−9.50T+53T2 1 - 9.50T + 53T^{2}
59 1−14.8T+59T2 1 - 14.8T + 59T^{2}
61 1+12.3T+61T2 1 + 12.3T + 61T^{2}
67 1+0.993T+67T2 1 + 0.993T + 67T^{2}
71 1−4.42T+71T2 1 - 4.42T + 71T^{2}
73 1−2.32T+73T2 1 - 2.32T + 73T^{2}
79 1+12.5T+79T2 1 + 12.5T + 79T^{2}
83 1−15.1T+83T2 1 - 15.1T + 83T^{2}
89 1+3.41T+89T2 1 + 3.41T + 89T^{2}
97 1+13.6T+97T2 1 + 13.6T + 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

−9.394211971808337714366431050190, −8.996571107349110890366579419615, −8.286399491738411070998189275903, −7.13217545175246791281468520635, −6.37382444802770303715476333651, −5.90606153591497433606478854091, −4.16182532945671102375721882002, −3.45923228931344902655259676374, −2.17501530240991508313784865739, −1.15363645918272117607549757472, 1.15363645918272117607549757472, 2.17501530240991508313784865739, 3.45923228931344902655259676374, 4.16182532945671102375721882002, 5.90606153591497433606478854091, 6.37382444802770303715476333651, 7.13217545175246791281468520635, 8.286399491738411070998189275903, 8.996571107349110890366579419615, 9.394211971808337714366431050190

Graph of the ZZ-function along the critical line