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Properties

Label 2-396-132.131-c0-0-2
Degree 22
Conductor 396396
Sign 0.816−0.577i0.816 - 0.577i
Analytic cond. 0.1976290.197629
Root an. cond. 0.4445550.444555
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s − 1.41i·5-s + 1.41·7-s − i·8-s + 1.41·10-s + i·11-s + 1.41i·14-s + 16-s − 1.41·19-s + 1.41i·20-s − 22-s − 1.00·25-s − 1.41·28-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s − 1.41i·5-s + 1.41·7-s − i·8-s + 1.41·10-s + i·11-s + 1.41i·14-s + 16-s − 1.41·19-s + 1.41i·20-s − 22-s − 1.00·25-s − 1.41·28-s + i·32-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=((0.816−0.577i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s) L(s)=((0.816−0.577i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: 0.816−0.577i0.816 - 0.577i
Analytic conductor: 0.1976290.197629
Root analytic conductor: 0.4445550.444555
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ396(395,⋅)\chi_{396} (395, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 396, ( :0), 0.816−0.577i)(2,\ 396,\ (\ :0),\ 0.816 - 0.577i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.83333331960.8333333196
L(12)L(\frac12) ≈\approx 0.83333331960.8333333196
L(1)L(1) 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−iT 1 - iT
3 1 1
11 1−iT 1 - iT
good5 1+1.41iT−T2 1 + 1.41iT - T^{2}
7 1−1.41T+T2 1 - 1.41T + T^{2}
13 1−T2 1 - T^{2}
17 1+T2 1 + T^{2}
19 1+1.41T+T2 1 + 1.41T + T^{2}
23 1+T2 1 + T^{2}
29 1+T2 1 + T^{2}
31 1−T2 1 - T^{2}
37 1+T2 1 + T^{2}
41 1+T2 1 + T^{2}
43 1+1.41T+T2 1 + 1.41T + T^{2}
47 1+T2 1 + T^{2}
53 1−1.41iT−T2 1 - 1.41iT - T^{2}
59 1+T2 1 + T^{2}
61 1−T2 1 - T^{2}
67 1−T2 1 - T^{2}
71 1+T2 1 + T^{2}
73 1−T2 1 - T^{2}
79 1+1.41T+T2 1 + 1.41T + T^{2}
83 1−T2 1 - T^{2}
89 1+1.41iT−T2 1 + 1.41iT - T^{2}
97 1+2T+T2 1 + 2T + T^{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

−11.84811778468000534321762618270, −10.53546698480487644901203954389, −9.433538405310197988787395072462, −8.552842018811519360573732951411, −8.087823647879042209377343137560, −7.03174875722134841349653682731, −5.69116736876525591781948201456, −4.72417463360781465426598235763, −4.34634277242828612828933908640, −1.61885775541892759896728718420, 1.93204183660956945955701069077, 3.09401800829321625355083712985, 4.23878676859564224235093214418, 5.46440498461878671844368034694, 6.69381307210043456248616850124, 8.035827601218306045933111181729, 8.607991993189091257825554573210, 9.978029709582538840840246299421, 10.87648308423117055710174489632, 11.12723351507651080270080549041

Graph of the ZZ-function along the critical line