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Properties

Label 2-396-12.11-c1-0-10
Degree 22
Conductor 396396
Sign 0.713−0.700i0.713 - 0.700i
Analytic cond. 3.162073.16207
Root an. cond. 1.778221.77822
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  + (1.40 + 0.126i)2-s + (1.96 + 0.355i)4-s + 0.450i·5-s + 4.60i·7-s + (2.72 + 0.749i)8-s + (−0.0568 + 0.635i)10-s + 11-s − 5.39·13-s + (−0.580 + 6.48i)14-s + (3.74 + 1.39i)16-s − 5.39i·17-s − 4.38i·19-s + (−0.160 + 0.887i)20-s + (1.40 + 0.126i)22-s + 6.03·23-s + ⋯
L(s)  = 1  + (0.996 + 0.0892i)2-s + (0.984 + 0.177i)4-s + 0.201i·5-s + 1.73i·7-s + (0.964 + 0.264i)8-s + (−0.0179 + 0.200i)10-s + 0.301·11-s − 1.49·13-s + (−0.155 + 1.73i)14-s + (0.936 + 0.349i)16-s − 1.30i·17-s − 1.00i·19-s + (−0.0358 + 0.198i)20-s + (0.300 + 0.0269i)22-s + 1.25·23-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=((0.713−0.700i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s+1/2) L(s)=((0.713−0.700i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: 0.713−0.700i0.713 - 0.700i
Analytic conductor: 3.162073.16207
Root analytic conductor: 1.778221.77822
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ396(287,⋅)\chi_{396} (287, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 396, ( :1/2), 0.713−0.700i)(2,\ 396,\ (\ :1/2),\ 0.713 - 0.700i)

Particular Values

L(1)L(1) ≈\approx 2.28581+0.935108i2.28581 + 0.935108i
L(12)L(\frac12) ≈\approx 2.28581+0.935108i2.28581 + 0.935108i
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+(−1.40−0.126i)T 1 + (-1.40 - 0.126i)T
3 1 1
11 1−T 1 - T
good5 1−0.450iT−5T2 1 - 0.450iT - 5T^{2}
7 1−4.60iT−7T2 1 - 4.60iT - 7T^{2}
13 1+5.39T+13T2 1 + 5.39T + 13T^{2}
17 1+5.39iT−17T2 1 + 5.39iT - 17T^{2}
19 1+4.38iT−19T2 1 + 4.38iT - 19T^{2}
23 1−6.03T+23T2 1 - 6.03T + 23T^{2}
29 1+3.31iT−29T2 1 + 3.31iT - 29T^{2}
31 1−4.57iT−31T2 1 - 4.57iT - 31T^{2}
37 1+6.27T+37T2 1 + 6.27T + 37T^{2}
41 1+11.2iT−41T2 1 + 11.2iT - 41T^{2}
43 1−0.650iT−43T2 1 - 0.650iT - 43T^{2}
47 1+5.79T+47T2 1 + 5.79T + 47T^{2}
53 1+1.46iT−53T2 1 + 1.46iT - 53T^{2}
59 1+4.71T+59T2 1 + 4.71T + 59T^{2}
61 1−1.28T+61T2 1 - 1.28T + 61T^{2}
67 1−6.54iT−67T2 1 - 6.54iT - 67T^{2}
71 1+4.27T+71T2 1 + 4.27T + 71T^{2}
73 1−9.86T+73T2 1 - 9.86T + 73T^{2}
79 1+8.84iT−79T2 1 + 8.84iT - 79T^{2}
83 1−0.627T+83T2 1 - 0.627T + 83T^{2}
89 1−10.6iT−89T2 1 - 10.6iT - 89T^{2}
97 1−1.59T+97T2 1 - 1.59T + 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

−11.78422280892013307963503863133, −10.80204686157535179213423947901, −9.489965416818222679094810786868, −8.727621305546165109267125126324, −7.29209903362016011315648734104, −6.64773471608812130153214171030, −5.24265742128131439296829156000, −4.94251591409866345626461134012, −3.03052273869441256492680415119, −2.36392490154396051482374813492, 1.44488150154958125321263598547, 3.24424111703717207463844384646, 4.25749939258322920799666565170, 5.06182519592477222176879684227, 6.47809166738915922944339829252, 7.21142537285468929186876549900, 8.067970422792474361705938569575, 9.724832200891843081922800786014, 10.47906997662161040172955740478, 11.14286191469737448469164666256

Graph of the ZZ-function along the critical line