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Properties

Label 2-74-1.1-c7-0-12
Degree 22
Conductor 7474
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 36.0·3-s + 64·4-s − 19.7·5-s + 288.·6-s − 50.9·7-s − 512·8-s − 886.·9-s + 157.·10-s + 5.17e3·11-s − 2.30e3·12-s + 4.31e3·13-s + 407.·14-s + 711.·15-s + 4.09e3·16-s + 1.40e4·17-s + 7.09e3·18-s + 2.66e4·19-s − 1.26e3·20-s + 1.83e3·21-s − 4.14e4·22-s − 7.74e4·23-s + 1.84e4·24-s − 7.77e4·25-s − 3.44e4·26-s + 1.10e5·27-s − 3.25e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.771·3-s + 0.5·4-s − 0.0706·5-s + 0.545·6-s − 0.0561·7-s − 0.353·8-s − 0.405·9-s + 0.0499·10-s + 1.17·11-s − 0.385·12-s + 0.544·13-s + 0.0396·14-s + 0.0544·15-s + 0.250·16-s + 0.694·17-s + 0.286·18-s + 0.891·19-s − 0.0353·20-s + 0.0432·21-s − 0.829·22-s − 1.32·23-s + 0.272·24-s − 0.995·25-s − 0.384·26-s + 1.08·27-s − 0.0280·28-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: −1-1
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: 11
Selberg data: (2, 74, ( :7/2), −1)(2,\ 74,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
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+36.0T+2.18e3T2 1 + 36.0T + 2.18e3T^{2}
5 1+19.7T+7.81e4T2 1 + 19.7T + 7.81e4T^{2}
7 1+50.9T+8.23e5T2 1 + 50.9T + 8.23e5T^{2}
11 1−5.17e3T+1.94e7T2 1 - 5.17e3T + 1.94e7T^{2}
13 1−4.31e3T+6.27e7T2 1 - 4.31e3T + 6.27e7T^{2}
17 1−1.40e4T+4.10e8T2 1 - 1.40e4T + 4.10e8T^{2}
19 1−2.66e4T+8.93e8T2 1 - 2.66e4T + 8.93e8T^{2}
23 1+7.74e4T+3.40e9T2 1 + 7.74e4T + 3.40e9T^{2}
29 1+3.31e4T+1.72e10T2 1 + 3.31e4T + 1.72e10T^{2}
31 1+1.72e5T+2.75e10T2 1 + 1.72e5T + 2.75e10T^{2}
41 1+8.46e5T+1.94e11T2 1 + 8.46e5T + 1.94e11T^{2}
43 1+3.07e5T+2.71e11T2 1 + 3.07e5T + 2.71e11T^{2}
47 1+3.22e5T+5.06e11T2 1 + 3.22e5T + 5.06e11T^{2}
53 1−1.83e6T+1.17e12T2 1 - 1.83e6T + 1.17e12T^{2}
59 1+7.97e5T+2.48e12T2 1 + 7.97e5T + 2.48e12T^{2}
61 1+6.84e5T+3.14e12T2 1 + 6.84e5T + 3.14e12T^{2}
67 1+3.79e6T+6.06e12T2 1 + 3.79e6T + 6.06e12T^{2}
71 1+3.42e5T+9.09e12T2 1 + 3.42e5T + 9.09e12T^{2}
73 1−2.64e6T+1.10e13T2 1 - 2.64e6T + 1.10e13T^{2}
79 1+2.44e6T+1.92e13T2 1 + 2.44e6T + 1.92e13T^{2}
83 1−5.61e6T+2.71e13T2 1 - 5.61e6T + 2.71e13T^{2}
89 1−8.43e6T+4.42e13T2 1 - 8.43e6T + 4.42e13T^{2}
97 1+3.40e6T+8.07e13T2 1 + 3.40e6T + 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

−11.97516636307086132533139138572, −11.61792304272230807525287413550, −10.32215851552829083112350510814, −9.221640083167962114200095471210, −7.975799993867005169480289000987, −6.56161858019835799145226513004, −5.55486254918469052860634700054, −3.58610015018173937621774895127, −1.48555689459506694529918385714, 0, 1.48555689459506694529918385714, 3.58610015018173937621774895127, 5.55486254918469052860634700054, 6.56161858019835799145226513004, 7.975799993867005169480289000987, 9.221640083167962114200095471210, 10.32215851552829083112350510814, 11.61792304272230807525287413550, 11.97516636307086132533139138572

Graph of the ZZ-function along the critical line