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Properties

Label 2-1320-5.4-c1-0-17
Degree 22
Conductor 13201320
Sign 0.998−0.0566i0.998 - 0.0566i
Analytic cond. 10.540210.5402
Root an. cond. 3.246573.24657
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·3-s + (2.23 − 0.126i)5-s + 3.92i·7-s − 9-s − 11-s − 2.60i·13-s + (−0.126 − 2.23i)15-s − 2.54i·17-s + 4.21·19-s + 3.92·21-s + 3.65i·23-s + (4.96 − 0.565i)25-s + i·27-s + 3.40·29-s + 9.61·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.998 − 0.0566i)5-s + 1.48i·7-s − 0.333·9-s − 0.301·11-s − 0.723i·13-s + (−0.0326 − 0.576i)15-s − 0.616i·17-s + 0.966·19-s + 0.856·21-s + 0.762i·23-s + (0.993 − 0.113i)25-s + 0.192i·27-s + 0.631·29-s + 1.72·31-s + ⋯

Functional equation

Λ(s)=(1320s/2 ΓC(s) L(s)=((0.998−0.0566i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0566i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1320s/2 ΓC(s+1/2) L(s)=((0.998−0.0566i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0566i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13201320    =    23⋅3⋅5⋅112^{3} \cdot 3 \cdot 5 \cdot 11
Sign: 0.998−0.0566i0.998 - 0.0566i
Analytic conductor: 10.540210.5402
Root analytic conductor: 3.246573.24657
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1320(529,⋅)\chi_{1320} (529, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1320, ( :1/2), 0.998−0.0566i)(2,\ 1320,\ (\ :1/2),\ 0.998 - 0.0566i)

Particular Values

L(1)L(1) ≈\approx 2.0138247112.013824711
L(12)L(\frac12) ≈\approx 2.0138247112.013824711
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+iT 1 + iT
5 1+(−2.23+0.126i)T 1 + (-2.23 + 0.126i)T
11 1+T 1 + T
good7 1−3.92iT−7T2 1 - 3.92iT - 7T^{2}
13 1+2.60iT−13T2 1 + 2.60iT - 13T^{2}
17 1+2.54iT−17T2 1 + 2.54iT - 17T^{2}
19 1−4.21T+19T2 1 - 4.21T + 19T^{2}
23 1−3.65iT−23T2 1 - 3.65iT - 23T^{2}
29 1−3.40T+29T2 1 - 3.40T + 29T^{2}
31 1−9.61T+31T2 1 - 9.61T + 31T^{2}
37 1−6.71iT−37T2 1 - 6.71iT - 37T^{2}
41 1+4.53T+41T2 1 + 4.53T + 41T^{2}
43 1+2.14iT−43T2 1 + 2.14iT - 43T^{2}
47 1−11.5iT−47T2 1 - 11.5iT - 47T^{2}
53 1+8.67iT−53T2 1 + 8.67iT - 53T^{2}
59 1−13.4T+59T2 1 - 13.4T + 59T^{2}
61 1−7.78T+61T2 1 - 7.78T + 61T^{2}
67 1−11.7iT−67T2 1 - 11.7iT - 67T^{2}
71 1+1.38T+71T2 1 + 1.38T + 71T^{2}
73 1−5.75iT−73T2 1 - 5.75iT - 73T^{2}
79 1+2.93T+79T2 1 + 2.93T + 79T^{2}
83 1+16.4iT−83T2 1 + 16.4iT - 83T^{2}
89 1+15.6T+89T2 1 + 15.6T + 89T^{2}
97 1+7.29iT−97T2 1 + 7.29iT - 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

−9.743045768422915568613318813672, −8.714278665513436324959575560299, −8.215058836178849497695674645175, −7.10379822866485175595653848203, −6.20804419935666646067876146811, −5.50899196929967628474438740208, −4.96616205915339478148728450095, −3.00941280973980306987487135668, −2.50848682678238450181739080292, −1.22403795735428968035805199733, 1.01464679219656282296837923192, 2.42216054001782342089826904279, 3.63945234530643998121188040464, 4.49402664709133296481132752198, 5.32351365341865917733483626986, 6.41719488195800304008597184975, 7.00945406811493526428105329665, 8.083151348584149043398392305116, 8.937055880380474458911304438130, 9.939144595814732887307770776854

Graph of the ZZ-function along the critical line