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Properties

Label 2-2600-104.3-c0-0-1
Degree 22
Conductor 26002600
Sign 0.711+0.702i0.711 + 0.702i
Analytic cond. 1.297561.29756
Root an. cond. 1.139101.13910
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s − 0.999i·18-s + (−0.5 + 0.866i)19-s + (0.866 + 0.499i)22-s + (−1.73 + i)23-s + 0.999·26-s + (0.866 − 0.499i)28-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s − 0.999i·18-s + (−0.5 + 0.866i)19-s + (0.866 + 0.499i)22-s + (−1.73 + i)23-s + 0.999·26-s + (0.866 − 0.499i)28-s + ⋯

Functional equation

Λ(s)=(2600s/2 ΓC(s) L(s)=((0.711+0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2600s/2 ΓC(s) L(s)=((0.711+0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26002600    =    23⋅52⋅132^{3} \cdot 5^{2} \cdot 13
Sign: 0.711+0.702i0.711 + 0.702i
Analytic conductor: 1.297561.29756
Root analytic conductor: 1.139101.13910
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2600(1251,⋅)\chi_{2600} (1251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2600, ( :0), 0.711+0.702i)(2,\ 2600,\ (\ :0),\ 0.711 + 0.702i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.3312976762.331297676
L(12)L(\frac12) ≈\approx 2.3312976762.331297676
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)
bad2 1+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
5 1 1
13 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
good3 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
7 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
11 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
17 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
19 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
23 1+(1.73−i)T+(0.5−0.866i)T2 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1−T2 1 - T^{2}
37 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
41 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
43 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
47 1+iT−T2 1 + iT - T^{2}
53 1−iT−T2 1 - iT - T^{2}
59 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+T2 1 + T^{2}
79 1−T2 1 - T^{2}
83 1+T2 1 + T^{2}
89 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
97 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
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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

−9.076642343936522377521252169828, −8.278268736623450508859377964752, −7.21425173183678276256614158767, −6.50124884871405497195740788601, −5.78855115778608661062616825562, −4.96196569180102712580734685827, −3.91627357165780003260548606954, −3.70145068503795363484559927788, −1.97187022198770110473106143615, −1.61018348605316155779429868423, 1.54290568411418966785320005331, 2.66582262675062126342870991697, 3.82692314081633048224677467433, 4.41293593820350935875989440071, 5.19048447538248347690322585920, 6.07930160130108524365927013176, 6.72119592626390885985633431552, 7.66141646773502031640132174288, 8.263060163704091803249581421363, 8.683523341662305151145158447384

Graph of the ZZ-function along the critical line