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Properties

Label 2-23-23.22-c18-0-30
Degree 22
Conductor 2323
Sign 11
Analytic cond. 47.238847.2388
Root an. cond. 6.873046.87304
Motivic weight 1818
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.00e3·2-s + 2.82e4·3-s + 7.39e5·4-s + 2.82e7·6-s + 4.78e8·8-s + 4.09e8·9-s + 2.08e10·12-s − 1.44e10·13-s + 2.84e11·16-s + 4.10e11·18-s − 1.80e12·23-s + 1.35e13·24-s + 3.81e12·25-s − 1.44e13·26-s + 6.30e11·27-s − 2.58e13·29-s + 4.63e13·31-s + 1.59e14·32-s + 3.03e14·36-s − 4.06e14·39-s − 5.38e14·41-s − 1.80e15·46-s − 2.01e15·47-s + 8.03e15·48-s + 1.62e15·49-s + 3.81e15·50-s − 1.06e16·52-s + ⋯
L(s)  = 1  + 1.95·2-s + 1.43·3-s + 2.82·4-s + 2.80·6-s + 3.56·8-s + 1.05·9-s + 4.04·12-s − 1.35·13-s + 4.14·16-s + 2.06·18-s − 23-s + 5.11·24-s + 25-s − 2.65·26-s + 0.0826·27-s − 1.77·29-s + 1.75·31-s + 4.53·32-s + 2.98·36-s − 1.94·39-s − 1.64·41-s − 1.95·46-s − 1.80·47-s + 5.94·48-s + 49-s + 1.95·50-s − 3.83·52-s + ⋯

Functional equation

Λ(s)=(23s/2 ΓC(s) L(s)=( Λ(19−s)\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}
Λ(s)=(23s/2 ΓC(s+9) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2323
Sign: 11
Analytic conductor: 47.238847.2388
Root analytic conductor: 6.873046.87304
Motivic weight: 1818
Rational: yes
Arithmetic: yes
Character: χ23(22,⋅)\chi_{23} (22, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 23, ( :9), 1)(2,\ 23,\ (\ :9),\ 1)

Particular Values

L(192)L(\frac{19}{2}) ≈\approx 11.6782291811.67822918
L(12)L(\frac12) ≈\approx 11.6782291811.67822918
L(10)L(10) 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+p9T 1 + p^{9} T
good2 1−1001T+p18T2 1 - 1001 T + p^{18} T^{2}
3 1−28234T+p18T2 1 - 28234 T + p^{18} T^{2}
5 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
7 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
11 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
13 1+14401098646T+p18T2 1 + 14401098646 T + p^{18} T^{2}
17 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
19 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
29 1+25800611881462T+p18T2 1 + 25800611881462 T + p^{18} T^{2}
31 1−46387544624642T+p18T2 1 - 46387544624642 T + p^{18} T^{2}
37 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
41 1+538722240191278T+p18T2 1 + 538722240191278 T + p^{18} T^{2}
43 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
47 1+2018037720599134T+p18T2 1 + 2018037720599134 T + p^{18} T^{2}
53 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
59 1−15758386442415578T+p18T2 1 - 15758386442415578 T + p^{18} T^{2}
61 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
67 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
71 1−40558698720275762T+p18T2 1 - 40558698720275762 T + p^{18} T^{2}
73 1−51973347295686674T+p18T2 1 - 51973347295686674 T + p^{18} T^{2}
79 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
83 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
89 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
97 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} 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

−13.83877100077117935979267215193, −12.88298105588427695720233511801, −11.73368804452336289856193641480, −10.00654669708963870958569000323, −7.986220614631621341268842343611, −6.82422588773599657434278828448, −5.13038915151909024959381179601, −3.88305664283450861490644797012, −2.81275167606681373192316097020, −1.94329165165678285053919402493, 1.94329165165678285053919402493, 2.81275167606681373192316097020, 3.88305664283450861490644797012, 5.13038915151909024959381179601, 6.82422588773599657434278828448, 7.986220614631621341268842343611, 10.00654669708963870958569000323, 11.73368804452336289856193641480, 12.88298105588427695720233511801, 13.83877100077117935979267215193

Graph of the ZZ-function along the critical line