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Properties

Label 2-3276-91.83-c1-0-45
Degree 22
Conductor 32763276
Sign −0.500+0.865i-0.500 + 0.865i
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  + (1.55 + 1.55i)5-s + (1.18 − 2.36i)7-s + (0.517 + 0.517i)11-s + (1.24 − 3.38i)13-s − 7.17·17-s + (−2.67 − 2.67i)19-s − 3.88i·23-s − 0.146i·25-s − 9.37·29-s + (1.62 + 1.62i)31-s + (5.53 − 1.83i)35-s + (−5.07 − 5.07i)37-s + (−6.07 − 6.07i)41-s − 5.25i·43-s + (−0.280 + 0.280i)47-s + ⋯
L(s)  = 1  + (0.696 + 0.696i)5-s + (0.448 − 0.894i)7-s + (0.156 + 0.156i)11-s + (0.345 − 0.938i)13-s − 1.74·17-s + (−0.614 − 0.614i)19-s − 0.810i·23-s − 0.0292i·25-s − 1.74·29-s + (0.291 + 0.291i)31-s + (0.934 − 0.310i)35-s + (−0.834 − 0.834i)37-s + (−0.948 − 0.948i)41-s − 0.801i·43-s + (−0.0408 + 0.0408i)47-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=((−0.500+0.865i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=((−0.500+0.865i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: −0.500+0.865i-0.500 + 0.865i
Analytic conductor: 26.158926.1589
Root analytic conductor: 5.114585.11458
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3276(2449,⋅)\chi_{3276} (2449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3276, ( :1/2), −0.500+0.865i)(2,\ 3276,\ (\ :1/2),\ -0.500 + 0.865i)

Particular Values

L(1)L(1) ≈\approx 1.2021813941.202181394
L(12)L(\frac12) ≈\approx 1.2021813941.202181394
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+(−1.18+2.36i)T 1 + (-1.18 + 2.36i)T
13 1+(−1.24+3.38i)T 1 + (-1.24 + 3.38i)T
good5 1+(−1.55−1.55i)T+5iT2 1 + (-1.55 - 1.55i)T + 5iT^{2}
11 1+(−0.517−0.517i)T+11iT2 1 + (-0.517 - 0.517i)T + 11iT^{2}
17 1+7.17T+17T2 1 + 7.17T + 17T^{2}
19 1+(2.67+2.67i)T+19iT2 1 + (2.67 + 2.67i)T + 19iT^{2}
23 1+3.88iT−23T2 1 + 3.88iT - 23T^{2}
29 1+9.37T+29T2 1 + 9.37T + 29T^{2}
31 1+(−1.62−1.62i)T+31iT2 1 + (-1.62 - 1.62i)T + 31iT^{2}
37 1+(5.07+5.07i)T+37iT2 1 + (5.07 + 5.07i)T + 37iT^{2}
41 1+(6.07+6.07i)T+41iT2 1 + (6.07 + 6.07i)T + 41iT^{2}
43 1+5.25iT−43T2 1 + 5.25iT - 43T^{2}
47 1+(0.280−0.280i)T−47iT2 1 + (0.280 - 0.280i)T - 47iT^{2}
53 1−8.34T+53T2 1 - 8.34T + 53T^{2}
59 1+(4.23−4.23i)T−59iT2 1 + (4.23 - 4.23i)T - 59iT^{2}
61 1−5.93iT−61T2 1 - 5.93iT - 61T^{2}
67 1+(8.85−8.85i)T−67iT2 1 + (8.85 - 8.85i)T - 67iT^{2}
71 1+(7.82−7.82i)T−71iT2 1 + (7.82 - 7.82i)T - 71iT^{2}
73 1+(−6.56+6.56i)T−73iT2 1 + (-6.56 + 6.56i)T - 73iT^{2}
79 1+14.3T+79T2 1 + 14.3T + 79T^{2}
83 1+(−7.45−7.45i)T+83iT2 1 + (-7.45 - 7.45i)T + 83iT^{2}
89 1+(1.55−1.55i)T−89iT2 1 + (1.55 - 1.55i)T - 89iT^{2}
97 1+(−7.33−7.33i)T+97iT2 1 + (-7.33 - 7.33i)T + 97iT^{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

−8.628224873260600305953794880022, −7.38771983985478505148365709620, −6.98918157224863431962675906523, −6.23084011461144620233414703810, −5.40311206467828710025409744643, −4.44191404134460072037175831798, −3.74220582108548194207589846624, −2.56597795083153808137310908896, −1.84003582986176350464656815048, −0.32581592800466845761802418816, 1.71498637101189785313466181428, 1.94283766000368905107725620686, 3.36839029929544958645493134287, 4.45167421163417593800647607234, 5.02914302834159960577353770682, 5.96959631828536645940354605301, 6.39055758665412101881722194132, 7.44101582560570339012444711525, 8.417632756502913698451823216587, 8.953486312788529670147696824461

Graph of the ZZ-function along the critical line