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Properties

Label 2-1805-95.59-c0-0-1
Degree 22
Conductor 18051805
Sign 0.934+0.356i0.934 + 0.356i
Analytic cond. 0.9008120.900812
Root an. cond. 0.9491110.949111
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  + (0.939 + 0.342i)4-s + (−0.939 + 0.342i)5-s + (−0.173 − 0.984i)9-s + (1 − 1.73i)11-s + (0.766 + 0.642i)16-s − 20-s + (0.766 − 0.642i)25-s + (0.173 − 0.984i)36-s + (1.53 − 1.28i)44-s + (0.5 + 0.866i)45-s + (−0.5 + 0.866i)49-s + (−0.347 + 1.96i)55-s + (1.87 + 0.684i)61-s + (0.500 + 0.866i)64-s + (−0.939 − 0.342i)80-s + (−0.939 + 0.342i)81-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)4-s + (−0.939 + 0.342i)5-s + (−0.173 − 0.984i)9-s + (1 − 1.73i)11-s + (0.766 + 0.642i)16-s − 20-s + (0.766 − 0.642i)25-s + (0.173 − 0.984i)36-s + (1.53 − 1.28i)44-s + (0.5 + 0.866i)45-s + (−0.5 + 0.866i)49-s + (−0.347 + 1.96i)55-s + (1.87 + 0.684i)61-s + (0.500 + 0.866i)64-s + (−0.939 − 0.342i)80-s + (−0.939 + 0.342i)81-s + ⋯

Functional equation

Λ(s)=(1805s/2 ΓC(s) L(s)=((0.934+0.356i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1805s/2 ΓC(s) L(s)=((0.934+0.356i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18051805    =    5⋅1925 \cdot 19^{2}
Sign: 0.934+0.356i0.934 + 0.356i
Analytic conductor: 0.9008120.900812
Root analytic conductor: 0.9491110.949111
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1805(1199,⋅)\chi_{1805} (1199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1805, ( :0), 0.934+0.356i)(2,\ 1805,\ (\ :0),\ 0.934 + 0.356i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2563261721.256326172
L(12)L(\frac12) ≈\approx 1.2563261721.256326172
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+(0.939−0.342i)T 1 + (0.939 - 0.342i)T
19 1 1
good2 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
3 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
7 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
13 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
17 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
23 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
29 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+T2 1 + T^{2}
41 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
43 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
47 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
53 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
59 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
61 1+(−1.87−0.684i)T+(0.766+0.642i)T2 1 + (-1.87 - 0.684i)T + (0.766 + 0.642i)T^{2}
67 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
71 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
73 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
79 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
83 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
89 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
97 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)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

−9.226866457780885609149043900824, −8.517439331773773822211621526116, −7.896049966513681869691897202696, −6.88128931310006562275676072200, −6.43804552585346934447718078479, −5.65428764236974543620038661643, −4.04883269614687562857412384411, −3.49275855474324674702688405565, −2.77956927167641606079338957195, −1.05990217004758937017361241634, 1.50995150883724228360282460664, 2.40970250396672848062314110143, 3.70558103905421531082912955313, 4.63244764492594870151805979057, 5.33750336412210329528675190446, 6.57056498811158570249991166592, 7.16732812334231693788432522025, 7.76959870443401389935655513046, 8.609225270588778728887401822860, 9.650385640120276749075800718007

Graph of the ZZ-function along the critical line