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Properties

Label 2-396-132.119-c1-0-11
Degree $2$
Conductor $396$
Sign $0.369 + 0.929i$
Analytic cond. $3.16207$
Root an. cond. $1.77822$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.399 − 1.35i)2-s + (−1.68 + 1.08i)4-s + (1.73 − 0.564i)5-s + (−1.29 + 1.77i)7-s + (2.14 + 1.84i)8-s + (−1.46 − 2.13i)10-s + (3.30 − 0.287i)11-s + (1.55 − 4.78i)13-s + (2.92 + 1.03i)14-s + (1.64 − 3.64i)16-s + (2.77 − 0.902i)17-s + (2.06 + 2.83i)19-s + (−2.30 + 2.83i)20-s + (−1.71 − 4.36i)22-s + 3.20·23-s + ⋯
L(s)  = 1  + (−0.282 − 0.959i)2-s + (−0.840 + 0.542i)4-s + (0.776 − 0.252i)5-s + (−0.487 + 0.671i)7-s + (0.757 + 0.652i)8-s + (−0.461 − 0.673i)10-s + (0.996 − 0.0866i)11-s + (0.430 − 1.32i)13-s + (0.781 + 0.277i)14-s + (0.411 − 0.911i)16-s + (0.673 − 0.218i)17-s + (0.472 + 0.650i)19-s + (−0.515 + 0.633i)20-s + (−0.364 − 0.931i)22-s + 0.668·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(396\)    =    \(2^{2} \cdot 3^{2} \cdot 11\)
Sign: $0.369 + 0.929i$
Analytic conductor: \(3.16207\)
Root analytic conductor: \(1.77822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{396} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 396,\ (\ :1/2),\ 0.369 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06040 - 0.719126i\)
\(L(\frac12)\) \(\approx\) \(1.06040 - 0.719126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.399 + 1.35i)T \)
3 \( 1 \)
11 \( 1 + (-3.30 + 0.287i)T \)
good5 \( 1 + (-1.73 + 0.564i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (1.29 - 1.77i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.55 + 4.78i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.77 + 0.902i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.06 - 2.83i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.20T + 23T^{2} \)
29 \( 1 + (-4.65 + 6.40i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (8.07 + 2.62i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.25 - 5.27i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.91 + 2.64i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.91iT - 43T^{2} \)
47 \( 1 + (-1.08 + 0.785i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.71 + 1.20i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.77 + 5.64i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.01 - 3.13i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 9.45iT - 67T^{2} \)
71 \( 1 + (-3.78 - 11.6i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.21 + 1.61i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.68 + 1.19i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.0446 + 0.137i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 11.8iT - 89T^{2} \)
97 \( 1 + (4.71 - 14.5i)T + (-78.4 - 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.14419258759416434227118833111, −9.992275465827379612764936833938, −9.552578661696340711634891039557, −8.668967114231525851443513793033, −7.68772046242962112599703784617, −6.07199680861533494360542209847, −5.32089263500376492390189186680, −3.76543149940769987266456612993, −2.71616902393866881534965411553, −1.21458775742403272756574514268, 1.41205807606029098408651674392, 3.60344751652727735462576208967, 4.76079491710804335249425396432, 6.07265208087497962727287179045, 6.71226488614521713059590251030, 7.46679420886760056011807444738, 8.969136265906014382439467705557, 9.358400861494975855695409710302, 10.29676573784713772540571927516, 11.23748548966641597632293472621

Graph of the $Z$-function along the critical line