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Properties

Label 2-2070-69.68-c1-0-23
Degree 22
Conductor 20702070
Sign −0.353+0.935i-0.353 + 0.935i
Analytic cond. 16.529016.5290
Root an. cond. 4.065594.06559
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s + 5-s + 0.891i·7-s + i·8-s − i·10-s + 0.713·11-s − 4.24·13-s + 0.891·14-s + 16-s + 0.522·17-s − 2.05i·19-s − 20-s − 0.713i·22-s + (−1.20 − 4.64i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.447·5-s + 0.337i·7-s + 0.353i·8-s − 0.316i·10-s + 0.214·11-s − 1.17·13-s + 0.238·14-s + 0.250·16-s + 0.126·17-s − 0.472i·19-s − 0.223·20-s − 0.152i·22-s + (−0.251 − 0.967i)23-s + ⋯

Functional equation

Λ(s)=(2070s/2 ΓC(s) L(s)=((−0.353+0.935i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2070s/2 ΓC(s+1/2) L(s)=((−0.353+0.935i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20702070    =    2⋅32⋅5⋅232 \cdot 3^{2} \cdot 5 \cdot 23
Sign: −0.353+0.935i-0.353 + 0.935i
Analytic conductor: 16.529016.5290
Root analytic conductor: 4.065594.06559
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2070(1241,⋅)\chi_{2070} (1241, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2070, ( :1/2), −0.353+0.935i)(2,\ 2070,\ (\ :1/2),\ -0.353 + 0.935i)

Particular Values

L(1)L(1) ≈\approx 1.4618420641.461842064
L(12)L(\frac12) ≈\approx 1.4618420641.461842064
L(32)L(\frac{3}{2}) 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+iT 1 + iT
3 1 1
5 1−T 1 - T
23 1+(1.20+4.64i)T 1 + (1.20 + 4.64i)T
good7 1−0.891iT−7T2 1 - 0.891iT - 7T^{2}
11 1−0.713T+11T2 1 - 0.713T + 11T^{2}
13 1+4.24T+13T2 1 + 4.24T + 13T^{2}
17 1−0.522T+17T2 1 - 0.522T + 17T^{2}
19 1+2.05iT−19T2 1 + 2.05iT - 19T^{2}
29 1+3.14iT−29T2 1 + 3.14iT - 29T^{2}
31 1−5.09T+31T2 1 - 5.09T + 31T^{2}
37 1+10.9iT−37T2 1 + 10.9iT - 37T^{2}
41 1−4.32iT−41T2 1 - 4.32iT - 41T^{2}
43 1+10.8iT−43T2 1 + 10.8iT - 43T^{2}
47 1−8.11iT−47T2 1 - 8.11iT - 47T^{2}
53 1−10.1T+53T2 1 - 10.1T + 53T^{2}
59 1+8.77iT−59T2 1 + 8.77iT - 59T^{2}
61 1+10.4iT−61T2 1 + 10.4iT - 61T^{2}
67 1−0.835iT−67T2 1 - 0.835iT - 67T^{2}
71 1−4.15iT−71T2 1 - 4.15iT - 71T^{2}
73 1−4.80T+73T2 1 - 4.80T + 73T^{2}
79 1+1.14iT−79T2 1 + 1.14iT - 79T^{2}
83 1−3.24T+83T2 1 - 3.24T + 83T^{2}
89 1−9.10T+89T2 1 - 9.10T + 89T^{2}
97 1−4.69iT−97T2 1 - 4.69iT - 97T^{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

−9.096905567708490342837467540276, −8.283085656218937715375615937217, −7.37146276540635901988395454822, −6.47132348952403288300501173348, −5.54585392898016310198533057193, −4.77926281016227415204067744657, −3.90585071546692231287113516211, −2.65027912824302139404340402146, −2.12471113582238875926675544135, −0.56442456790118267468628352210, 1.20932110791408262184393672017, 2.58099990397584892951241560374, 3.72689911318784957490701393560, 4.70835688659607482173428799921, 5.41512620926806882998835148166, 6.26545523863871338523628216823, 7.05403681595689018405494572643, 7.67619892098869959421407212714, 8.507530756510602475428626217107, 9.338498161231691444373806341089

Graph of the ZZ-function along the critical line