↑

Properties

Label 2-23-23.4-c1-0-0
Degree 22
Conductor 2323
Sign 0.960−0.278i0.960 - 0.278i
Analytic cond. 0.1836550.183655
Root an. cond. 0.4285500.428550
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.198 + 0.435i)2-s + (−2.11 + 0.620i)3-s + (1.15 − 1.33i)4-s + (−2.18 − 1.40i)5-s + (−0.691 − 0.797i)6-s + (0.483 + 3.36i)7-s + (1.73 + 0.508i)8-s + (1.56 − 1.00i)9-s + (0.176 − 1.22i)10-s + (0.0950 − 0.208i)11-s + (−1.62 + 3.54i)12-s + (0.435 − 3.02i)13-s + (−1.36 + 0.879i)14-s + (5.48 + 1.61i)15-s + (−0.380 − 2.64i)16-s + (1.26 + 1.45i)17-s + ⋯
L(s)  = 1  + (0.140 + 0.308i)2-s + (−1.22 + 0.358i)3-s + (0.579 − 0.669i)4-s + (−0.976 − 0.627i)5-s + (−0.282 − 0.325i)6-s + (0.182 + 1.27i)7-s + (0.612 + 0.179i)8-s + (0.520 − 0.334i)9-s + (0.0559 − 0.388i)10-s + (0.0286 − 0.0627i)11-s + (−0.467 + 1.02i)12-s + (0.120 − 0.839i)13-s + (−0.365 + 0.235i)14-s + (1.41 + 0.415i)15-s + (−0.0952 − 0.662i)16-s + (0.306 + 0.353i)17-s + ⋯

Functional equation

Λ(s)=(23s/2 ΓC(s) L(s)=((0.960−0.278i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(23s/2 ΓC(s+1/2) L(s)=((0.960−0.278i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2323
Sign: 0.960−0.278i0.960 - 0.278i
Analytic conductor: 0.1836550.183655
Root analytic conductor: 0.4285500.428550
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ23(4,⋅)\chi_{23} (4, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 23, ( :1/2), 0.960−0.278i)(2,\ 23,\ (\ :1/2),\ 0.960 - 0.278i)

Particular Values

L(1)L(1) ≈\approx 0.522645+0.0741564i0.522645 + 0.0741564i
L(12)L(\frac12) ≈\approx 0.522645+0.0741564i0.522645 + 0.0741564i
L(32)L(\frac{3}{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)
bad23 1+(4.62+1.25i)T 1 + (4.62 + 1.25i)T
good2 1+(−0.198−0.435i)T+(−1.30+1.51i)T2 1 + (-0.198 - 0.435i)T + (-1.30 + 1.51i)T^{2}
3 1+(2.11−0.620i)T+(2.52−1.62i)T2 1 + (2.11 - 0.620i)T + (2.52 - 1.62i)T^{2}
5 1+(2.18+1.40i)T+(2.07+4.54i)T2 1 + (2.18 + 1.40i)T + (2.07 + 4.54i)T^{2}
7 1+(−0.483−3.36i)T+(−6.71+1.97i)T2 1 + (-0.483 - 3.36i)T + (-6.71 + 1.97i)T^{2}
11 1+(−0.0950+0.208i)T+(−7.20−8.31i)T2 1 + (-0.0950 + 0.208i)T + (-7.20 - 8.31i)T^{2}
13 1+(−0.435+3.02i)T+(−12.4−3.66i)T2 1 + (-0.435 + 3.02i)T + (-12.4 - 3.66i)T^{2}
17 1+(−1.26−1.45i)T+(−2.41+16.8i)T2 1 + (-1.26 - 1.45i)T + (-2.41 + 16.8i)T^{2}
19 1+(1.26−1.46i)T+(−2.70−18.8i)T2 1 + (1.26 - 1.46i)T + (-2.70 - 18.8i)T^{2}
29 1+(−4.23−4.89i)T+(−4.12+28.7i)T2 1 + (-4.23 - 4.89i)T + (-4.12 + 28.7i)T^{2}
31 1+(1.44+0.424i)T+(26.0+16.7i)T2 1 + (1.44 + 0.424i)T + (26.0 + 16.7i)T^{2}
37 1+(5.67−3.64i)T+(15.3−33.6i)T2 1 + (5.67 - 3.64i)T + (15.3 - 33.6i)T^{2}
41 1+(−6.78−4.36i)T+(17.0+37.2i)T2 1 + (-6.78 - 4.36i)T + (17.0 + 37.2i)T^{2}
43 1+(2.55−0.749i)T+(36.1−23.2i)T2 1 + (2.55 - 0.749i)T + (36.1 - 23.2i)T^{2}
47 1+1.43T+47T2 1 + 1.43T + 47T^{2}
53 1+(1.22+8.49i)T+(−50.8+14.9i)T2 1 + (1.22 + 8.49i)T + (-50.8 + 14.9i)T^{2}
59 1+(0.00878−0.0611i)T+(−56.6−16.6i)T2 1 + (0.00878 - 0.0611i)T + (-56.6 - 16.6i)T^{2}
61 1+(−0.0426−0.0125i)T+(51.3+32.9i)T2 1 + (-0.0426 - 0.0125i)T + (51.3 + 32.9i)T^{2}
67 1+(−5.15−11.2i)T+(−43.8+50.6i)T2 1 + (-5.15 - 11.2i)T + (-43.8 + 50.6i)T^{2}
71 1+(3.46+7.58i)T+(−46.4+53.6i)T2 1 + (3.46 + 7.58i)T + (-46.4 + 53.6i)T^{2}
73 1+(−0.437+0.505i)T+(−10.3−72.2i)T2 1 + (-0.437 + 0.505i)T + (-10.3 - 72.2i)T^{2}
79 1+(−1.70+11.8i)T+(−75.7−22.2i)T2 1 + (-1.70 + 11.8i)T + (-75.7 - 22.2i)T^{2}
83 1+(0.303−0.194i)T+(34.4−75.4i)T2 1 + (0.303 - 0.194i)T + (34.4 - 75.4i)T^{2}
89 1+(−15.4+4.54i)T+(74.8−48.1i)T2 1 + (-15.4 + 4.54i)T + (74.8 - 48.1i)T^{2}
97 1+(0.335+0.215i)T+(40.2+88.2i)T2 1 + (0.335 + 0.215i)T + (40.2 + 88.2i)T^{2}
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

−17.86050679187521477445319340182, −16.34346305871690337678584639273, −15.82622556652173074562931536094, −14.76213691508780106542578001162, −12.37142508458322437423924466078, −11.60364304372605875531843957745, −10.37030895406267864126384531529, −8.245259285711677999907873202935, −6.09176650110013088563725642668, −5.02391178006753994249697831566, 4.02744855059760795335469833124, 6.71610085194056859926149333211, 7.62411840852442737605105286610, 10.66371080744931333057356507937, 11.43344959459257525705111511349, 12.27355814871927646413260670117, 13.91452657831320887142581353274, 15.80233741763860238681677209725, 16.77549438280340543921186921433, 17.67124850837853196870804823785

Graph of the ZZ-function along the critical line