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Properties

Label 2-4002-1.1-c1-0-12
Degree 22
Conductor 40024002
Sign 11
Analytic cond. 31.956131.9561
Root an. cond. 5.652975.65297
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 + 0.230·5-s + 6-s + 4.59·7-s − 8-s + 9-s − 0.230·10-s − 4.50·11-s − 12-s + 1.15·13-s − 4.59·14-s − 0.230·15-s + 16-s − 7.42·17-s − 18-s − 5.94·19-s + 0.230·20-s − 4.59·21-s + 4.50·22-s + 23-s + 24-s − 4.94·25-s − 1.15·26-s − 27-s + 4.59·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.102·5-s + 0.408·6-s + 1.73·7-s − 0.353·8-s + 0.333·9-s − 0.0728·10-s − 1.35·11-s − 0.288·12-s + 0.320·13-s − 1.22·14-s − 0.0594·15-s + 0.250·16-s − 1.80·17-s − 0.235·18-s − 1.36·19-s + 0.0514·20-s − 1.00·21-s + 0.960·22-s + 0.208·23-s + 0.204·24-s − 0.989·25-s − 0.226·26-s − 0.192·27-s + 0.868·28-s + ⋯

Functional equation

Λ(s)=(4002s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4002s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40024002    =    2⋅3⋅23⋅292 \cdot 3 \cdot 23 \cdot 29
Sign: 11
Analytic conductor: 31.956131.9561
Root analytic conductor: 5.652975.65297
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4002, ( :1/2), 1)(2,\ 4002,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0480993661.048099366
L(12)L(\frac12) ≈\approx 1.0480993661.048099366
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
23 1−T 1 - T
29 1+T 1 + T
good5 1−0.230T+5T2 1 - 0.230T + 5T^{2}
7 1−4.59T+7T2 1 - 4.59T + 7T^{2}
11 1+4.50T+11T2 1 + 4.50T + 11T^{2}
13 1−1.15T+13T2 1 - 1.15T + 13T^{2}
17 1+7.42T+17T2 1 + 7.42T + 17T^{2}
19 1+5.94T+19T2 1 + 5.94T + 19T^{2}
31 1−6.50T+31T2 1 - 6.50T + 31T^{2}
37 1−7.29T+37T2 1 - 7.29T + 37T^{2}
41 1−0.0953T+41T2 1 - 0.0953T + 41T^{2}
43 1−0.941T+43T2 1 - 0.941T + 43T^{2}
47 1+0.598T+47T2 1 + 0.598T + 47T^{2}
53 1−13.3T+53T2 1 - 13.3T + 53T^{2}
59 1+2.94T+59T2 1 + 2.94T + 59T^{2}
61 1−10.8T+61T2 1 - 10.8T + 61T^{2}
67 1−2.09T+67T2 1 - 2.09T + 67T^{2}
71 1−10.9T+71T2 1 - 10.9T + 71T^{2}
73 1+12.3T+73T2 1 + 12.3T + 73T^{2}
79 1−16.5T+79T2 1 - 16.5T + 79T^{2}
83 1+1.71T+83T2 1 + 1.71T + 83T^{2}
89 1−12.5T+89T2 1 - 12.5T + 89T^{2}
97 1+1.82T+97T2 1 + 1.82T + 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

−8.308141225689569048871619558886, −7.958710481808238266256809122228, −7.09062906218052413076981248971, −6.28636572405820789937375183650, −5.50863859880605912001720531561, −4.70674940525600118228113621584, −4.14820613937935349855855121250, −2.38882215644980644418351265159, −2.01502943840024056540401026618, −0.65810877460570271866895200525, 0.65810877460570271866895200525, 2.01502943840024056540401026618, 2.38882215644980644418351265159, 4.14820613937935349855855121250, 4.70674940525600118228113621584, 5.50863859880605912001720531561, 6.28636572405820789937375183650, 7.09062906218052413076981248971, 7.958710481808238266256809122228, 8.308141225689569048871619558886

Graph of the ZZ-function along the critical line