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Properties

Label 2-2695-2695.989-c0-0-3
Degree 22
Conductor 26952695
Sign −0.999−0.0427i-0.999 - 0.0427i
Analytic cond. 1.344981.34498
Root an. cond. 1.159731.15973
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  + (−1.46 − 1.35i)2-s + (0.222 + 2.96i)4-s + (−0.365 − 0.930i)5-s + (0.974 − 0.222i)7-s + (2.45 − 3.08i)8-s + (0.955 + 0.294i)9-s + (−0.728 + 1.85i)10-s + (−0.955 + 0.294i)11-s + (0.131 − 0.574i)13-s + (−1.72 − 0.997i)14-s + (−4.83 + 0.728i)16-s + (−1.29 − 0.880i)17-s + (−0.997 − 1.72i)18-s + (2.68 − 1.29i)20-s + (1.79 + 0.865i)22-s + ⋯
L(s)  = 1  + (−1.46 − 1.35i)2-s + (0.222 + 2.96i)4-s + (−0.365 − 0.930i)5-s + (0.974 − 0.222i)7-s + (2.45 − 3.08i)8-s + (0.955 + 0.294i)9-s + (−0.728 + 1.85i)10-s + (−0.955 + 0.294i)11-s + (0.131 − 0.574i)13-s + (−1.72 − 0.997i)14-s + (−4.83 + 0.728i)16-s + (−1.29 − 0.880i)17-s + (−0.997 − 1.72i)18-s + (2.68 − 1.29i)20-s + (1.79 + 0.865i)22-s + ⋯

Functional equation

Λ(s)=(2695s/2 ΓC(s) L(s)=((−0.999−0.0427i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0427i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2695s/2 ΓC(s) L(s)=((−0.999−0.0427i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0427i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26952695    =    5⋅72⋅115 \cdot 7^{2} \cdot 11
Sign: −0.999−0.0427i-0.999 - 0.0427i
Analytic conductor: 1.344981.34498
Root analytic conductor: 1.159731.15973
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2695(989,⋅)\chi_{2695} (989, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2695, ( :0), −0.999−0.0427i)(2,\ 2695,\ (\ :0),\ -0.999 - 0.0427i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.46085995220.4608599522
L(12)L(\frac12) ≈\approx 0.46085995220.4608599522
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)
bad5 1+(0.365+0.930i)T 1 + (0.365 + 0.930i)T
7 1+(−0.974+0.222i)T 1 + (-0.974 + 0.222i)T
11 1+(0.955−0.294i)T 1 + (0.955 - 0.294i)T
good2 1+(1.46+1.35i)T+(0.0747+0.997i)T2 1 + (1.46 + 1.35i)T + (0.0747 + 0.997i)T^{2}
3 1+(−0.955−0.294i)T2 1 + (-0.955 - 0.294i)T^{2}
13 1+(−0.131+0.574i)T+(−0.900−0.433i)T2 1 + (-0.131 + 0.574i)T + (-0.900 - 0.433i)T^{2}
17 1+(1.29+0.880i)T+(0.365+0.930i)T2 1 + (1.29 + 0.880i)T + (0.365 + 0.930i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(−0.365+0.930i)T2 1 + (-0.365 + 0.930i)T^{2}
29 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
31 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
37 1+(0.988+0.149i)T2 1 + (0.988 + 0.149i)T^{2}
41 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
43 1+(0.848+1.06i)T+(−0.222+0.974i)T2 1 + (0.848 + 1.06i)T + (-0.222 + 0.974i)T^{2}
47 1+(−0.0747−0.997i)T2 1 + (-0.0747 - 0.997i)T^{2}
53 1+(0.988−0.149i)T2 1 + (0.988 - 0.149i)T^{2}
59 1+(−0.698+1.77i)T+(−0.733−0.680i)T2 1 + (-0.698 + 1.77i)T + (-0.733 - 0.680i)T^{2}
61 1+(0.988+0.149i)T2 1 + (0.988 + 0.149i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1+(0.134+0.0648i)T+(0.623+0.781i)T2 1 + (0.134 + 0.0648i)T + (0.623 + 0.781i)T^{2}
73 1+(0.218−0.202i)T+(0.0747−0.997i)T2 1 + (0.218 - 0.202i)T + (0.0747 - 0.997i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+(0.385+1.68i)T+(−0.900+0.433i)T2 1 + (0.385 + 1.68i)T + (-0.900 + 0.433i)T^{2}
89 1+(−1.40−0.432i)T+(0.826+0.563i)T2 1 + (-1.40 - 0.432i)T + (0.826 + 0.563i)T^{2}
97 1−T2 1 - 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

−8.717773005538867618088674222005, −8.078575599378339396369175324636, −7.59132520341507262272280744509, −6.97986014348315848697568218591, −5.05189737490408896542096342273, −4.48927545386794456644851242116, −3.66163276322341407091501170582, −2.34782010463142967799150504031, −1.70514066582846541918155187286, −0.50055968809830891596922625533, 1.46450815457367865343857866523, 2.35325555972764351483042499730, 4.20238872736099066646764394851, 4.99280711236762921201177810659, 5.96143033302811143258588506103, 6.67467110769857406111841859948, 7.20631311632882000695368066452, 7.904120954415321436256724971740, 8.475335364938012260808306453224, 9.130272294774618139451082753040

Graph of the ZZ-function along the critical line