↑

Properties

Label 2-23-23.22-c30-0-37
Degree 22
Conductor 2323
Sign 11
Analytic cond. 131.132131.132
Root an. cond. 11.451311.4513
Motivic weight 3030
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.04e4·2-s + 1.97e7·3-s + 1.46e9·4-s − 9.98e11·6-s − 1.98e13·8-s + 1.86e14·9-s + 2.90e16·12-s + 9.69e16·13-s − 5.75e17·16-s − 9.38e18·18-s − 2.66e20·23-s − 3.92e20·24-s + 9.31e20·25-s − 4.88e21·26-s − 3.91e20·27-s − 4.57e21·29-s + 3.15e22·31-s + 5.03e22·32-s + 2.73e23·36-s + 1.91e24·39-s + 3.10e24·41-s + 1.34e25·46-s − 1.77e25·47-s − 1.14e25·48-s + 2.25e25·49-s − 4.69e25·50-s + 1.42e26·52-s + ⋯
L(s)  = 1  − 1.53·2-s + 1.37·3-s + 1.36·4-s − 2.12·6-s − 0.563·8-s + 0.904·9-s + 1.88·12-s + 1.89·13-s − 0.499·16-s − 1.39·18-s − 23-s − 0.777·24-s + 25-s − 2.91·26-s − 0.132·27-s − 0.530·29-s + 1.34·31-s + 1.33·32-s + 1.23·36-s + 2.61·39-s + 1.99·41-s + 1.53·46-s − 1.46·47-s − 0.689·48-s + 49-s − 1.53·50-s + 2.58·52-s + ⋯

Functional equation

Λ(s)=(23s/2 ΓC(s) L(s)=( Λ(31−s)\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(31-s) \end{aligned}
Λ(s)=(23s/2 ΓC(s+15) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2323
Sign: 11
Analytic conductor: 131.132131.132
Root analytic conductor: 11.451311.4513
Motivic weight: 3030
Rational: yes
Arithmetic: yes
Character: χ23(22,⋅)\chi_{23} (22, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 23, ( :15), 1)(2,\ 23,\ (\ :15),\ 1)

Particular Values

L(312)L(\frac{31}{2}) ≈\approx 1.9813366161.981336616
L(12)L(\frac12) ≈\approx 1.9813366161.981336616
L(16)L(16) 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+p15T 1 + p^{15} T
good2 1+50407T+p30T2 1 + 50407 T + p^{30} T^{2}
3 1−19799482T+p30T2 1 - 19799482 T + p^{30} T^{2}
5 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
7 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
11 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
13 1−96954465195037082T+p30T2 1 - 96954465195037082 T + p^{30} T^{2}
17 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
19 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
29 1+ 1 + 45 ⁣⋯ ⁣7445\!\cdots\!74T+p30T2 T + p^{30} T^{2}
31 1− 1 - 31 ⁣⋯ ⁣7431\!\cdots\!74T+p30T2 T + p^{30} T^{2}
37 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
41 1− 1 - 31 ⁣⋯ ⁣7431\!\cdots\!74T+p30T2 T + p^{30} T^{2}
43 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
47 1+ 1 + 17 ⁣⋯ ⁣8217\!\cdots\!82T+p30T2 T + p^{30} T^{2}
53 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
59 1− 1 - 13 ⁣⋯ ⁣9813\!\cdots\!98T+p30T2 T + p^{30} T^{2}
61 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
67 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
71 1+ 1 + 32 ⁣⋯ ⁣2632\!\cdots\!26T+p30T2 T + p^{30} T^{2}
73 1+ 1 + 50 ⁣⋯ ⁣1850\!\cdots\!18T+p30T2 T + p^{30} T^{2}
79 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
83 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
89 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
97 (1−p15T)(1+p15T) ( 1 - p^{15} T )( 1 + p^{15} T )
show more
show less
   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

−11.32179138845266023916613259846, −10.20636451449380484021649492062, −9.081666179065423840549007780265, −8.464331808217308924425201463959, −7.65872226315270299905087521141, −6.27255152329906035879746329481, −4.04866669702370114220443149468, −2.82950479287640721629202862451, −1.72565852388529464991066952301, −0.795779337645979707375269002458, 0.795779337645979707375269002458, 1.72565852388529464991066952301, 2.82950479287640721629202862451, 4.04866669702370114220443149468, 6.27255152329906035879746329481, 7.65872226315270299905087521141, 8.464331808217308924425201463959, 9.081666179065423840549007780265, 10.20636451449380484021649492062, 11.32179138845266023916613259846

Graph of the ZZ-function along the critical line