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Properties

Label 2-310-1.1-c7-0-22
Degree 22
Conductor 310310
Sign −1-1
Analytic cond. 96.839396.8393
Root an. cond. 9.840699.84069
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 − 89.7·3-s + 64·4-s − 125·5-s + 718.·6-s + 78.5·7-s − 512·8-s + 5.87e3·9-s + 1.00e3·10-s − 509.·11-s − 5.74e3·12-s − 8.64e3·13-s − 628.·14-s + 1.12e4·15-s + 4.09e3·16-s − 3.24e4·17-s − 4.69e4·18-s − 2.09e4·19-s − 8.00e3·20-s − 7.04e3·21-s + 4.07e3·22-s + 2.52e4·23-s + 4.59e4·24-s + 1.56e4·25-s + 6.91e4·26-s − 3.31e5·27-s + 5.02e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.91·3-s + 0.5·4-s − 0.447·5-s + 1.35·6-s + 0.0865·7-s − 0.353·8-s + 2.68·9-s + 0.316·10-s − 0.115·11-s − 0.959·12-s − 1.09·13-s − 0.0611·14-s + 0.858·15-s + 0.250·16-s − 1.59·17-s − 1.89·18-s − 0.699·19-s − 0.223·20-s − 0.166·21-s + 0.0815·22-s + 0.432·23-s + 0.678·24-s + 0.199·25-s + 0.771·26-s − 3.23·27-s + 0.0432·28-s + ⋯

Functional equation

Λ(s)=(310s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(310s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 310310    =    2⋅5⋅312 \cdot 5 \cdot 31
Sign: −1-1
Analytic conductor: 96.839396.8393
Root analytic conductor: 9.840699.84069
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 310, ( :7/2), −1)(2,\ 310,\ (\ :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
5 1+125T 1 + 125T
31 1−2.97e4T 1 - 2.97e4T
good3 1+89.7T+2.18e3T2 1 + 89.7T + 2.18e3T^{2}
7 1−78.5T+8.23e5T2 1 - 78.5T + 8.23e5T^{2}
11 1+509.T+1.94e7T2 1 + 509.T + 1.94e7T^{2}
13 1+8.64e3T+6.27e7T2 1 + 8.64e3T + 6.27e7T^{2}
17 1+3.24e4T+4.10e8T2 1 + 3.24e4T + 4.10e8T^{2}
19 1+2.09e4T+8.93e8T2 1 + 2.09e4T + 8.93e8T^{2}
23 1−2.52e4T+3.40e9T2 1 - 2.52e4T + 3.40e9T^{2}
29 1−7.69e4T+1.72e10T2 1 - 7.69e4T + 1.72e10T^{2}
37 1−3.70e5T+9.49e10T2 1 - 3.70e5T + 9.49e10T^{2}
41 1−3.11e5T+1.94e11T2 1 - 3.11e5T + 1.94e11T^{2}
43 1−2.22e5T+2.71e11T2 1 - 2.22e5T + 2.71e11T^{2}
47 1+1.25e5T+5.06e11T2 1 + 1.25e5T + 5.06e11T^{2}
53 1+5.94e5T+1.17e12T2 1 + 5.94e5T + 1.17e12T^{2}
59 1−9.63e5T+2.48e12T2 1 - 9.63e5T + 2.48e12T^{2}
61 1−1.74e6T+3.14e12T2 1 - 1.74e6T + 3.14e12T^{2}
67 1−7.92e5T+6.06e12T2 1 - 7.92e5T + 6.06e12T^{2}
71 1−1.44e6T+9.09e12T2 1 - 1.44e6T + 9.09e12T^{2}
73 1−1.84e6T+1.10e13T2 1 - 1.84e6T + 1.10e13T^{2}
79 1−1.93e6T+1.92e13T2 1 - 1.93e6T + 1.92e13T^{2}
83 1−2.97e6T+2.71e13T2 1 - 2.97e6T + 2.71e13T^{2}
89 1−2.59e6T+4.42e13T2 1 - 2.59e6T + 4.42e13T^{2}
97 1−3.19e5T+8.07e13T2 1 - 3.19e5T + 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

−10.24402056713349332176983794792, −9.300123120005368297074454736180, −7.907550831078642799956146648500, −6.90289331496643545131931934246, −6.31738976340749762866149444332, −5.04706430408812849206071296041, −4.29746999832155717623803290732, −2.23362009122687125139139122543, −0.798499402436378994240514405169, 0, 0.798499402436378994240514405169, 2.23362009122687125139139122543, 4.29746999832155717623803290732, 5.04706430408812849206071296041, 6.31738976340749762866149444332, 6.90289331496643545131931934246, 7.907550831078642799956146648500, 9.300123120005368297074454736180, 10.24402056713349332176983794792

Graph of the ZZ-function along the critical line