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Properties

Label 2-4650-1.1-c1-0-36
Degree 22
Conductor 46504650
Sign 11
Analytic cond. 37.130437.1304
Root an. cond. 6.093476.09347
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 − 6-s − 0.133·7-s + 8-s + 9-s + 2.92·11-s − 12-s + 4.92·13-s − 0.133·14-s + 16-s + 3.13·17-s + 18-s − 0.266·19-s + 0.133·21-s + 2.92·22-s + 2·23-s − 24-s + 4.92·26-s − 27-s − 0.133·28-s + 3.13·29-s − 31-s + 32-s − 2.92·33-s + 3.13·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.0504·7-s + 0.353·8-s + 0.333·9-s + 0.881·11-s − 0.288·12-s + 1.36·13-s − 0.0356·14-s + 0.250·16-s + 0.759·17-s + 0.235·18-s − 0.0612·19-s + 0.0291·21-s + 0.623·22-s + 0.417·23-s − 0.204·24-s + 0.965·26-s − 0.192·27-s − 0.0252·28-s + 0.581·29-s − 0.179·31-s + 0.176·32-s − 0.509·33-s + 0.537·34-s + ⋯

Functional equation

Λ(s)=(4650s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4650s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46504650    =    2⋅3⋅52⋅312 \cdot 3 \cdot 5^{2} \cdot 31
Sign: 11
Analytic conductor: 37.130437.1304
Root analytic conductor: 6.093476.09347
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4650, ( :1/2), 1)(2,\ 4650,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0564009913.056400991
L(12)L(\frac12) ≈\approx 3.0564009913.056400991
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
5 1 1
31 1+T 1 + T
good7 1+0.133T+7T2 1 + 0.133T + 7T^{2}
11 1−2.92T+11T2 1 - 2.92T + 11T^{2}
13 1−4.92T+13T2 1 - 4.92T + 13T^{2}
17 1−3.13T+17T2 1 - 3.13T + 17T^{2}
19 1+0.266T+19T2 1 + 0.266T + 19T^{2}
23 1−2T+23T2 1 - 2T + 23T^{2}
29 1−3.13T+29T2 1 - 3.13T + 29T^{2}
37 1−0.924T+37T2 1 - 0.924T + 37T^{2}
41 1+11.9T+41T2 1 + 11.9T + 41T^{2}
43 1+2.05T+43T2 1 + 2.05T + 43T^{2}
47 1+6.84T+47T2 1 + 6.84T + 47T^{2}
53 1−11.1T+53T2 1 - 11.1T + 53T^{2}
59 1−9.84T+59T2 1 - 9.84T + 59T^{2}
61 1−4.92T+61T2 1 - 4.92T + 61T^{2}
67 1+7.84T+67T2 1 + 7.84T + 67T^{2}
71 1+15.1T+71T2 1 + 15.1T + 71T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1+1.13T+79T2 1 + 1.13T + 79T^{2}
83 1−13.1T+83T2 1 - 13.1T + 83T^{2}
89 1+10.4T+89T2 1 + 10.4T + 89T^{2}
97 1−8.71T+97T2 1 - 8.71T + 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

−8.343758761426842555047786304184, −7.33752787064830191040773383989, −6.60832806854210189179211412793, −6.15444031687452024189322047399, −5.37521498960997554902038239874, −4.65489861837693091810961611592, −3.72958304866677299021993123494, −3.24861434774606724455858577579, −1.82721525927059732864633649858, −0.967172360582310006964082412142, 0.967172360582310006964082412142, 1.82721525927059732864633649858, 3.24861434774606724455858577579, 3.72958304866677299021993123494, 4.65489861837693091810961611592, 5.37521498960997554902038239874, 6.15444031687452024189322047399, 6.60832806854210189179211412793, 7.33752787064830191040773383989, 8.343758761426842555047786304184

Graph of the ZZ-function along the critical line