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Properties

Label 2-23-23.22-c6-0-3
Degree 22
Conductor 2323
Sign 11
Analytic cond. 5.291245.29124
Root an. cond. 2.300272.30027
Motivic weight 66
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7·2-s − 38·3-s − 15·4-s + 266·6-s + 553·8-s + 715·9-s + 570·12-s + 1.08e3·13-s − 2.91e3·16-s − 5.00e3·18-s − 1.21e4·23-s − 2.10e4·24-s + 1.56e4·25-s − 7.57e3·26-s + 532·27-s + 3.07e4·29-s + 5.87e4·31-s − 1.50e4·32-s − 1.07e4·36-s − 4.11e4·39-s + 4.36e4·41-s + 8.51e4·46-s − 2.05e5·47-s + 1.10e5·48-s + 1.17e5·49-s − 1.09e5·50-s − 1.62e4·52-s + ⋯
L(s)  = 1  − 7/8·2-s − 1.40·3-s − 0.234·4-s + 1.23·6-s + 1.08·8-s + 0.980·9-s + 0.329·12-s + 0.492·13-s − 0.710·16-s − 0.858·18-s − 23-s − 1.52·24-s + 25-s − 0.430·26-s + 0.0270·27-s + 1.26·29-s + 1.97·31-s − 0.458·32-s − 0.229·36-s − 0.693·39-s + 0.633·41-s + 7/8·46-s − 1.97·47-s + 1.00·48-s + 49-s − 7/8·50-s − 0.115·52-s + ⋯

Functional equation

Λ(s)=(23s/2 ΓC(s) L(s)=( Λ(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}
Λ(s)=(23s/2 ΓC(s+3) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2323
Sign: 11
Analytic conductor: 5.291245.29124
Root analytic conductor: 2.300272.30027
Motivic weight: 66
Rational: yes
Arithmetic: yes
Character: χ23(22,⋅)\chi_{23} (22, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 23, ( :3), 1)(2,\ 23,\ (\ :3),\ 1)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.48803031510.4880303151
L(12)L(\frac12) ≈\approx 0.48803031510.4880303151
L(4)L(4) 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)
bad23 1+p3T 1 + p^{3} T
good2 1+7T+p6T2 1 + 7 T + p^{6} T^{2}
3 1+38T+p6T2 1 + 38 T + p^{6} T^{2}
5 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
7 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
11 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
13 1−1082T+p6T2 1 - 1082 T + p^{6} T^{2}
17 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
19 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
29 1−30746T+p6T2 1 - 30746 T + p^{6} T^{2}
31 1−58754T+p6T2 1 - 58754 T + p^{6} T^{2}
37 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
41 1−43634T+p6T2 1 - 43634 T + p^{6} T^{2}
43 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
47 1+205342T+p6T2 1 + 205342 T + p^{6} T^{2}
53 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
59 1+253942T+p6T2 1 + 253942 T + p^{6} T^{2}
61 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
67 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
71 1−667154T+p6T2 1 - 667154 T + p^{6} T^{2}
73 1−725042T+p6T2 1 - 725042 T + p^{6} T^{2}
79 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
83 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
89 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
97 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
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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

−16.85679340562610487422197990914, −15.89194571733631421976646400420, −13.85616011152672178118138225480, −12.31744179707429698461728489589, −10.99805223113660726529032535557, −9.963420897852036337490473396685, −8.290094947978572616300124743303, −6.44575988406637144103056549585, −4.74761605237732684069464542721, −0.815114495854622862890383705685, 0.815114495854622862890383705685, 4.74761605237732684069464542721, 6.44575988406637144103056549585, 8.290094947978572616300124743303, 9.963420897852036337490473396685, 10.99805223113660726529032535557, 12.31744179707429698461728489589, 13.85616011152672178118138225480, 15.89194571733631421976646400420, 16.85679340562610487422197990914

Graph of the ZZ-function along the critical line