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Properties

Label 2-1305-1.1-c3-0-53
Degree 22
Conductor 13051305
Sign 11
Analytic cond. 76.997476.9974
Root an. cond. 8.774828.77482
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.57·2-s − 1.37·4-s + 5·5-s + 31.5·7-s − 24.1·8-s + 12.8·10-s − 64.9·11-s + 26.2·13-s + 81.1·14-s − 51.0·16-s + 31.1·17-s − 75.4·19-s − 6.88·20-s − 167.·22-s + 207.·23-s + 25·25-s + 67.6·26-s − 43.4·28-s − 29·29-s + 269.·31-s + 61.5·32-s + 80.1·34-s + 157.·35-s + 254.·37-s − 194.·38-s − 120.·40-s − 65.6·41-s + ⋯
L(s)  = 1  + 0.909·2-s − 0.172·4-s + 0.447·5-s + 1.70·7-s − 1.06·8-s + 0.406·10-s − 1.77·11-s + 0.560·13-s + 1.54·14-s − 0.798·16-s + 0.444·17-s − 0.910·19-s − 0.0769·20-s − 1.61·22-s + 1.88·23-s + 0.200·25-s + 0.510·26-s − 0.293·28-s − 0.185·29-s + 1.56·31-s + 0.340·32-s + 0.404·34-s + 0.761·35-s + 1.13·37-s − 0.828·38-s − 0.476·40-s − 0.250·41-s + ⋯

Functional equation

Λ(s)=(1305s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1305s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13051305    =    32⋅5⋅293^{2} \cdot 5 \cdot 29
Sign: 11
Analytic conductor: 76.997476.9974
Root analytic conductor: 8.774828.77482
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1305, ( :3/2), 1)(2,\ 1305,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.6962375043.696237504
L(12)L(\frac12) ≈\approx 3.6962375043.696237504
L(52)L(\frac{5}{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)
bad3 1 1
5 1−5T 1 - 5T
29 1+29T 1 + 29T
good2 1−2.57T+8T2 1 - 2.57T + 8T^{2}
7 1−31.5T+343T2 1 - 31.5T + 343T^{2}
11 1+64.9T+1.33e3T2 1 + 64.9T + 1.33e3T^{2}
13 1−26.2T+2.19e3T2 1 - 26.2T + 2.19e3T^{2}
17 1−31.1T+4.91e3T2 1 - 31.1T + 4.91e3T^{2}
19 1+75.4T+6.85e3T2 1 + 75.4T + 6.85e3T^{2}
23 1−207.T+1.21e4T2 1 - 207.T + 1.21e4T^{2}
31 1−269.T+2.97e4T2 1 - 269.T + 2.97e4T^{2}
37 1−254.T+5.06e4T2 1 - 254.T + 5.06e4T^{2}
41 1+65.6T+6.89e4T2 1 + 65.6T + 6.89e4T^{2}
43 1−28.4T+7.95e4T2 1 - 28.4T + 7.95e4T^{2}
47 1+489.T+1.03e5T2 1 + 489.T + 1.03e5T^{2}
53 1−426.T+1.48e5T2 1 - 426.T + 1.48e5T^{2}
59 1+424.T+2.05e5T2 1 + 424.T + 2.05e5T^{2}
61 1+256.T+2.26e5T2 1 + 256.T + 2.26e5T^{2}
67 1−233.T+3.00e5T2 1 - 233.T + 3.00e5T^{2}
71 1−1.15e3T+3.57e5T2 1 - 1.15e3T + 3.57e5T^{2}
73 1+243.T+3.89e5T2 1 + 243.T + 3.89e5T^{2}
79 1−875.T+4.93e5T2 1 - 875.T + 4.93e5T^{2}
83 1−102.T+5.71e5T2 1 - 102.T + 5.71e5T^{2}
89 1−481.T+7.04e5T2 1 - 481.T + 7.04e5T^{2}
97 1+760.T+9.12e5T2 1 + 760.T + 9.12e5T^{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

−9.158678097131981669530840095442, −8.291510429467101266873622760336, −7.87604468143608413348385484002, −6.56837110283297938339529639550, −5.55924020381539532435242672893, −4.98370350037198139186335636093, −4.48836606616365249344251319079, −3.11957154074524617611821767225, −2.23571143552221040748030346376, −0.865463109453529623537002492798, 0.865463109453529623537002492798, 2.23571143552221040748030346376, 3.11957154074524617611821767225, 4.48836606616365249344251319079, 4.98370350037198139186335636093, 5.55924020381539532435242672893, 6.56837110283297938339529639550, 7.87604468143608413348385484002, 8.291510429467101266873622760336, 9.158678097131981669530840095442

Graph of the ZZ-function along the critical line