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Properties

Label 2-74-1.1-c7-0-3
Degree 22
Conductor 7474
Sign 11
Analytic cond. 23.116423.1164
Root an. cond. 4.807964.80796
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 53.5·3-s + 64·4-s − 314.·5-s − 428.·6-s − 1.16e3·7-s + 512·8-s + 675.·9-s − 2.51e3·10-s + 3.43e3·11-s − 3.42e3·12-s + 1.09e4·13-s − 9.31e3·14-s + 1.68e4·15-s + 4.09e3·16-s − 6.91e3·17-s + 5.40e3·18-s + 3.52e3·19-s − 2.01e4·20-s + 6.22e4·21-s + 2.74e4·22-s − 5.43e4·23-s − 2.73e4·24-s + 2.10e4·25-s + 8.73e4·26-s + 8.08e4·27-s − 7.45e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.14·3-s + 0.5·4-s − 1.12·5-s − 0.809·6-s − 1.28·7-s + 0.353·8-s + 0.309·9-s − 0.796·10-s + 0.778·11-s − 0.572·12-s + 1.37·13-s − 0.907·14-s + 1.28·15-s + 0.250·16-s − 0.341·17-s + 0.218·18-s + 0.117·19-s − 0.563·20-s + 1.46·21-s + 0.550·22-s − 0.931·23-s − 0.404·24-s + 0.269·25-s + 0.974·26-s + 0.790·27-s − 0.641·28-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: 11
Analytic conductor: 23.116423.1164
Root analytic conductor: 4.807964.80796
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 74, ( :7/2), 1)(2,\ 74,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 1.2355321851.235532185
L(12)L(\frac12) ≈\approx 1.2355321851.235532185
L(92)L(\frac{9}{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 1−8T 1 - 8T
37 1−5.06e4T 1 - 5.06e4T
good3 1+53.5T+2.18e3T2 1 + 53.5T + 2.18e3T^{2}
5 1+314.T+7.81e4T2 1 + 314.T + 7.81e4T^{2}
7 1+1.16e3T+8.23e5T2 1 + 1.16e3T + 8.23e5T^{2}
11 1−3.43e3T+1.94e7T2 1 - 3.43e3T + 1.94e7T^{2}
13 1−1.09e4T+6.27e7T2 1 - 1.09e4T + 6.27e7T^{2}
17 1+6.91e3T+4.10e8T2 1 + 6.91e3T + 4.10e8T^{2}
19 1−3.52e3T+8.93e8T2 1 - 3.52e3T + 8.93e8T^{2}
23 1+5.43e4T+3.40e9T2 1 + 5.43e4T + 3.40e9T^{2}
29 1−1.58e5T+1.72e10T2 1 - 1.58e5T + 1.72e10T^{2}
31 1−2.93e5T+2.75e10T2 1 - 2.93e5T + 2.75e10T^{2}
41 1−4.79e5T+1.94e11T2 1 - 4.79e5T + 1.94e11T^{2}
43 1+6.00e5T+2.71e11T2 1 + 6.00e5T + 2.71e11T^{2}
47 1+8.32e5T+5.06e11T2 1 + 8.32e5T + 5.06e11T^{2}
53 1−3.81e5T+1.17e12T2 1 - 3.81e5T + 1.17e12T^{2}
59 1−1.37e6T+2.48e12T2 1 - 1.37e6T + 2.48e12T^{2}
61 1+1.85e6T+3.14e12T2 1 + 1.85e6T + 3.14e12T^{2}
67 1−3.98e6T+6.06e12T2 1 - 3.98e6T + 6.06e12T^{2}
71 1−3.66e5T+9.09e12T2 1 - 3.66e5T + 9.09e12T^{2}
73 1−1.57e6T+1.10e13T2 1 - 1.57e6T + 1.10e13T^{2}
79 1+3.09e6T+1.92e13T2 1 + 3.09e6T + 1.92e13T^{2}
83 1+7.25e6T+2.71e13T2 1 + 7.25e6T + 2.71e13T^{2}
89 1+1.06e6T+4.42e13T2 1 + 1.06e6T + 4.42e13T^{2}
97 1−1.53e7T+8.07e13T2 1 - 1.53e7T + 8.07e13T^{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

−12.89913779677346345309491485500, −11.88320978913946296517962023016, −11.39717956886834694264403059226, −10.12233235668272731383308318420, −8.372338857126345129548462308586, −6.63877062187775463206450503666, −6.10526396147929208030885755746, −4.39751728168748794323009539412, −3.32615097875466929924972285720, −0.69880199121176732574555336500, 0.69880199121176732574555336500, 3.32615097875466929924972285720, 4.39751728168748794323009539412, 6.10526396147929208030885755746, 6.63877062187775463206450503666, 8.372338857126345129548462308586, 10.12233235668272731383308318420, 11.39717956886834694264403059226, 11.88320978913946296517962023016, 12.89913779677346345309491485500

Graph of the ZZ-function along the critical line