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Properties

Label 2-3276-13.12-c1-0-35
Degree 22
Conductor 32763276
Sign −0.978−0.205i-0.978 - 0.205i
Analytic cond. 26.158926.1589
Root an. cond. 5.114585.11458
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  − 0.257i·5-s − i·7-s − 4.19i·11-s + (0.742 − 3.52i)13-s − 7.46·17-s − 6.33i·19-s − 3.93·23-s + 4.93·25-s − 3.63·29-s + 8.55i·31-s − 0.257·35-s + 5.43i·37-s + 8.29i·41-s − 6.23·43-s + 9.91i·47-s + ⋯
L(s)  = 1  − 0.115i·5-s − 0.377i·7-s − 1.26i·11-s + (0.205 − 0.978i)13-s − 1.80·17-s − 1.45i·19-s − 0.820·23-s + 0.986·25-s − 0.674·29-s + 1.53i·31-s − 0.0435·35-s + 0.892i·37-s + 1.29i·41-s − 0.950·43-s + 1.44i·47-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=((−0.978−0.205i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=((−0.978−0.205i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: −0.978−0.205i-0.978 - 0.205i
Analytic conductor: 26.158926.1589
Root analytic conductor: 5.114585.11458
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3276(2521,⋅)\chi_{3276} (2521, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3276, ( :1/2), −0.978−0.205i)(2,\ 3276,\ (\ :1/2),\ -0.978 - 0.205i)

Particular Values

L(1)L(1) ≈\approx 0.42667927020.4266792702
L(12)L(\frac12) ≈\approx 0.42667927020.4266792702
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 1
3 1 1
7 1+iT 1 + iT
13 1+(−0.742+3.52i)T 1 + (-0.742 + 3.52i)T
good5 1+0.257iT−5T2 1 + 0.257iT - 5T^{2}
11 1+4.19iT−11T2 1 + 4.19iT - 11T^{2}
17 1+7.46T+17T2 1 + 7.46T + 17T^{2}
19 1+6.33iT−19T2 1 + 6.33iT - 19T^{2}
23 1+3.93T+23T2 1 + 3.93T + 23T^{2}
29 1+3.63T+29T2 1 + 3.63T + 29T^{2}
31 1−8.55iT−31T2 1 - 8.55iT - 31T^{2}
37 1−5.43iT−37T2 1 - 5.43iT - 37T^{2}
41 1−8.29iT−41T2 1 - 8.29iT - 41T^{2}
43 1+6.23T+43T2 1 + 6.23T + 43T^{2}
47 1−9.91iT−47T2 1 - 9.91iT - 47T^{2}
53 1+1.63T+53T2 1 + 1.63T + 53T^{2}
59 1−2.72iT−59T2 1 - 2.72iT - 59T^{2}
61 1+4.54T+61T2 1 + 4.54T + 61T^{2}
67 1+5.84iT−67T2 1 + 5.84iT - 67T^{2}
71 1+2.05iT−71T2 1 + 2.05iT - 71T^{2}
73 1−13.9iT−73T2 1 - 13.9iT - 73T^{2}
79 1+0.0979T+79T2 1 + 0.0979T + 79T^{2}
83 1+10.3iT−83T2 1 + 10.3iT - 83T^{2}
89 1+4.60iT−89T2 1 + 4.60iT - 89T^{2}
97 1+10.7iT−97T2 1 + 10.7iT - 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

−8.513965979552877327849077797193, −7.51008144219460891398072993009, −6.63740739323350107229419175433, −6.14330176090052074560079530896, −5.04926989913477320103263758639, −4.49767788954929080250343359117, −3.32089168728891205471997290263, −2.72928974954659336799779295943, −1.28610722456293395051990019899, −0.12690236858152335507266055452, 1.91519043399298284939048364015, 2.20056182311185154659689413356, 3.78467211047888398190177503111, 4.28296241601645308377711384704, 5.21075288437793978838093397227, 6.14977817920149072800704397075, 6.78986228817700037702550278460, 7.47594763328134768990965321619, 8.357440473233158882143378668977, 9.092505845228372312571938449566

Graph of the ZZ-function along the critical line