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Properties

Label 2-1260-1260.319-c0-0-2
Degree 22
Conductor 12601260
Sign −0.400+0.916i-0.400 + 0.916i
Analytic cond. 0.6288210.628821
Root an. cond. 0.7929820.792982
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  + (0.5 − 0.866i)2-s − 3-s + (−0.499 − 0.866i)4-s + 5-s + (−0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s − 0.999·8-s + 9-s + (0.5 − 0.866i)10-s + (0.499 + 0.866i)12-s + (−0.499 − 0.866i)14-s − 15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)18-s + (−0.499 − 0.866i)20-s + (−0.5 + 0.866i)21-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s − 3-s + (−0.499 − 0.866i)4-s + 5-s + (−0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s − 0.999·8-s + 9-s + (0.5 − 0.866i)10-s + (0.499 + 0.866i)12-s + (−0.499 − 0.866i)14-s − 15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)18-s + (−0.499 − 0.866i)20-s + (−0.5 + 0.866i)21-s + ⋯

Functional equation

Λ(s)=(1260s/2 ΓC(s) L(s)=((−0.400+0.916i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1260s/2 ΓC(s) L(s)=((−0.400+0.916i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12601260    =    22⋅32⋅5⋅72^{2} \cdot 3^{2} \cdot 5 \cdot 7
Sign: −0.400+0.916i-0.400 + 0.916i
Analytic conductor: 0.6288210.628821
Root analytic conductor: 0.7929820.792982
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1260(319,⋅)\chi_{1260} (319, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1260, ( :0), −0.400+0.916i)(2,\ 1260,\ (\ :0),\ -0.400 + 0.916i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1584821801.158482180
L(12)L(\frac12) ≈\approx 1.1584821801.158482180
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
3 1+T 1 + T
5 1−T 1 - T
7 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good11 1−T2 1 - T^{2}
13 1+(0.5+0.866i)T2 1 + (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)T2 1 + (0.5 - 0.866i)T^{2}
23 1−T+T2 1 - T + T^{2}
29 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(0.5−0.866i)T2 1 + (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)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
53 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)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

−9.877842846919813235236943117042, −9.324244686142360333497034116317, −7.985637042320783740292450537461, −6.79041848317970311452044166939, −6.14124722348907347234077433588, −5.18148788038742473475327454338, −4.65874982858307898869923180944, −3.57193518369719423967751122731, −2.10277314183847177237462616857, −1.08066306694577311942223332024, 1.78854771127628967216023435466, 3.23488698070796818721714174924, 4.66054248793872953131305007481, 5.34260740557641570116647952362, 5.75331704399695372293962118306, 6.73110307643248338198472121730, 7.30330046935198458234703319073, 8.613625797559795280706455998796, 9.125033515723090958650605778923, 10.10565276950189096421681690271

Graph of the ZZ-function along the critical line