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Properties

Label 2-1472-16.13-c1-0-6
Degree 22
Conductor 14721472
Sign −0.972+0.233i-0.972 + 0.233i
Analytic cond. 11.753911.7539
Root an. cond. 3.428403.42840
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  + (1.70 + 1.70i)3-s + (−2.61 + 2.61i)5-s − 1.66i·7-s + 2.81i·9-s + (−3.35 + 3.35i)11-s + (1.50 + 1.50i)13-s − 8.91·15-s − 0.812·17-s + (3.81 + 3.81i)19-s + (2.83 − 2.83i)21-s + i·23-s − 8.64i·25-s + (0.307 − 0.307i)27-s + (−7.12 − 7.12i)29-s − 10.7·31-s + ⋯
L(s)  = 1  + (0.984 + 0.984i)3-s + (−1.16 + 1.16i)5-s − 0.628i·7-s + 0.939i·9-s + (−1.01 + 1.01i)11-s + (0.418 + 0.418i)13-s − 2.30·15-s − 0.196·17-s + (0.876 + 0.876i)19-s + (0.618 − 0.618i)21-s + 0.208i·23-s − 1.72i·25-s + (0.0591 − 0.0591i)27-s + (−1.32 − 1.32i)29-s − 1.92·31-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=((−0.972+0.233i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s+1/2) L(s)=((−0.972+0.233i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: −0.972+0.233i-0.972 + 0.233i
Analytic conductor: 11.753911.7539
Root analytic conductor: 3.428403.42840
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1472(1105,⋅)\chi_{1472} (1105, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1472, ( :1/2), −0.972+0.233i)(2,\ 1472,\ (\ :1/2),\ -0.972 + 0.233i)

Particular Values

L(1)L(1) ≈\approx 1.0569851641.056985164
L(12)L(\frac12) ≈\approx 1.0569851641.056985164
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
23 1−iT 1 - iT
good3 1+(−1.70−1.70i)T+3iT2 1 + (-1.70 - 1.70i)T + 3iT^{2}
5 1+(2.61−2.61i)T−5iT2 1 + (2.61 - 2.61i)T - 5iT^{2}
7 1+1.66iT−7T2 1 + 1.66iT - 7T^{2}
11 1+(3.35−3.35i)T−11iT2 1 + (3.35 - 3.35i)T - 11iT^{2}
13 1+(−1.50−1.50i)T+13iT2 1 + (-1.50 - 1.50i)T + 13iT^{2}
17 1+0.812T+17T2 1 + 0.812T + 17T^{2}
19 1+(−3.81−3.81i)T+19iT2 1 + (-3.81 - 3.81i)T + 19iT^{2}
29 1+(7.12+7.12i)T+29iT2 1 + (7.12 + 7.12i)T + 29iT^{2}
31 1+10.7T+31T2 1 + 10.7T + 31T^{2}
37 1+(−3.80+3.80i)T−37iT2 1 + (-3.80 + 3.80i)T - 37iT^{2}
41 1+0.765iT−41T2 1 + 0.765iT - 41T^{2}
43 1+(3.75−3.75i)T−43iT2 1 + (3.75 - 3.75i)T - 43iT^{2}
47 1+2.18T+47T2 1 + 2.18T + 47T^{2}
53 1+(1.48−1.48i)T−53iT2 1 + (1.48 - 1.48i)T - 53iT^{2}
59 1+(9.46−9.46i)T−59iT2 1 + (9.46 - 9.46i)T - 59iT^{2}
61 1+(−0.442−0.442i)T+61iT2 1 + (-0.442 - 0.442i)T + 61iT^{2}
67 1+(−7.97−7.97i)T+67iT2 1 + (-7.97 - 7.97i)T + 67iT^{2}
71 1−2.88iT−71T2 1 - 2.88iT - 71T^{2}
73 1+7.28iT−73T2 1 + 7.28iT - 73T^{2}
79 1+1.36T+79T2 1 + 1.36T + 79T^{2}
83 1+(6.09+6.09i)T+83iT2 1 + (6.09 + 6.09i)T + 83iT^{2}
89 1−4.98iT−89T2 1 - 4.98iT - 89T^{2}
97 1+7.46T+97T2 1 + 7.46T + 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

−9.891378788878660018433069377975, −9.324936086670422878121236538456, −8.170172085805824562404391001484, −7.53277798681361663611466804839, −7.17225774410390131614827715130, −5.77314237660522884851775501797, −4.45248527815574227023222262254, −3.86160231954417103554387337780, −3.26901833931389464762193692457, −2.19705680088730852899014877330, 0.35943461661528111822623406829, 1.63936963922318427194120845881, 2.96166038542315889153484866352, 3.59322272641916278295060255289, 5.01705453403118562511552389832, 5.58622615137232815089757324156, 7.00893884023223518389403778888, 7.71521767945720023507613177080, 8.270949548885206774934914665835, 8.794157732099754003060855946518

Graph of the ZZ-function along the critical line