↑

Properties

Label 2-2303-2303.939-c0-0-4
Degree 22
Conductor 23032303
Sign 0.582+0.812i0.582 + 0.812i
Analytic cond. 1.149341.14934
Root an. cond. 1.072071.07207
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.385 + 0.483i)2-s + (0.708 − 0.341i)3-s + (0.137 − 0.602i)4-s + (0.437 + 0.210i)6-s + (−0.550 − 0.834i)7-s + (0.900 − 0.433i)8-s + (−0.238 + 0.298i)9-s + (−0.108 − 0.473i)12-s + (0.190 − 0.587i)14-s + (−0.437 − 1.91i)17-s − 0.236·18-s + (−0.674 − 0.403i)21-s + (0.490 − 0.614i)24-s + (0.623 − 0.781i)25-s + (−0.241 + 1.05i)27-s + (−0.578 + 0.217i)28-s + ⋯
L(s)  = 1  + (0.385 + 0.483i)2-s + (0.708 − 0.341i)3-s + (0.137 − 0.602i)4-s + (0.437 + 0.210i)6-s + (−0.550 − 0.834i)7-s + (0.900 − 0.433i)8-s + (−0.238 + 0.298i)9-s + (−0.108 − 0.473i)12-s + (0.190 − 0.587i)14-s + (−0.437 − 1.91i)17-s − 0.236·18-s + (−0.674 − 0.403i)21-s + (0.490 − 0.614i)24-s + (0.623 − 0.781i)25-s + (−0.241 + 1.05i)27-s + (−0.578 + 0.217i)28-s + ⋯

Functional equation

Λ(s)=(2303s/2 ΓC(s) L(s)=((0.582+0.812i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2303s/2 ΓC(s) L(s)=((0.582+0.812i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23032303    =    72⋅477^{2} \cdot 47
Sign: 0.582+0.812i0.582 + 0.812i
Analytic conductor: 1.149341.14934
Root analytic conductor: 1.072071.07207
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2303(939,⋅)\chi_{2303} (939, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2303, ( :0), 0.582+0.812i)(2,\ 2303,\ (\ :0),\ 0.582 + 0.812i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7627900371.762790037
L(12)L(\frac12) ≈\approx 1.7627900371.762790037
L(1)L(1) 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)
bad7 1+(0.550+0.834i)T 1 + (0.550 + 0.834i)T
47 1+(−0.623−0.781i)T 1 + (-0.623 - 0.781i)T
good2 1+(−0.385−0.483i)T+(−0.222+0.974i)T2 1 + (-0.385 - 0.483i)T + (-0.222 + 0.974i)T^{2}
3 1+(−0.708+0.341i)T+(0.623−0.781i)T2 1 + (-0.708 + 0.341i)T + (0.623 - 0.781i)T^{2}
5 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
11 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
13 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
17 1+(0.437+1.91i)T+(−0.900+0.433i)T2 1 + (0.437 + 1.91i)T + (-0.900 + 0.433i)T^{2}
19 1−T2 1 - T^{2}
23 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
29 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
31 1−T2 1 - T^{2}
37 1+(−0.429−1.87i)T+(−0.900+0.433i)T2 1 + (-0.429 - 1.87i)T + (-0.900 + 0.433i)T^{2}
41 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
43 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
53 1+(0.382−1.67i)T+(−0.900−0.433i)T2 1 + (0.382 - 1.67i)T + (-0.900 - 0.433i)T^{2}
59 1+(−1.24−0.599i)T+(0.623+0.781i)T2 1 + (-1.24 - 0.599i)T + (0.623 + 0.781i)T^{2}
61 1+(0.416+1.82i)T+(−0.900+0.433i)T2 1 + (0.416 + 1.82i)T + (-0.900 + 0.433i)T^{2}
67 1−T2 1 - T^{2}
71 1+(−0.174+0.766i)T+(−0.900−0.433i)T2 1 + (-0.174 + 0.766i)T + (-0.900 - 0.433i)T^{2}
73 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
79 1−0.947T+T2 1 - 0.947T + T^{2}
83 1+(0.277−0.347i)T+(−0.222−0.974i)T2 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2}
89 1+(−0.385+0.483i)T+(−0.222−0.974i)T2 1 + (-0.385 + 0.483i)T + (-0.222 - 0.974i)T^{2}
97 1−0.618T+T2 1 - 0.618T + 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

−9.152609853406232742107727603890, −8.080807149102222040025702266315, −7.39415848666228967288461796149, −6.81842095028109360437194408407, −6.12364731473370043704973415363, −4.99718922773366558875485119500, −4.49122203718087769088710745735, −3.19680946232453016208377440169, −2.40392279372308042359341831534, −1.00423766740494491729012477061, 1.97194515513467837255925467401, 2.69619479397757732159364510172, 3.65190466843018323311321176815, 3.98758122759132627073675584944, 5.28810086975675907860435298883, 6.13716963849216008039524924649, 7.00677322438659286275154354628, 8.031700284358141728320270060571, 8.661683946213194744827706028966, 9.090889299717543807047256839645

Graph of the ZZ-function along the critical line