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Properties

Label 2-396-12.11-c1-0-14
Degree $2$
Conductor $396$
Sign $0.918 + 0.396i$
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  + (0.382 + 1.36i)2-s + (−1.70 + 1.04i)4-s − 2.05i·5-s − 4.18i·7-s + (−2.07 − 1.92i)8-s + (2.79 − 0.785i)10-s − 11-s + 3.27·13-s + (5.70 − 1.60i)14-s + (1.83 − 3.55i)16-s + 0.332i·17-s − 8.47i·19-s + (2.13 + 3.50i)20-s + (−0.382 − 1.36i)22-s − 3.52·23-s + ⋯
L(s)  = 1  + (0.270 + 0.962i)2-s + (−0.853 + 0.520i)4-s − 0.918i·5-s − 1.58i·7-s + (−0.732 − 0.681i)8-s + (0.884 − 0.248i)10-s − 0.301·11-s + 0.908·13-s + (1.52 − 0.428i)14-s + (0.457 − 0.888i)16-s + 0.0806i·17-s − 1.94i·19-s + (0.478 + 0.784i)20-s + (−0.0815 − 0.290i)22-s − 0.734·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(396\)    =    \(2^{2} \cdot 3^{2} \cdot 11\)
Sign: $0.918 + 0.396i$
Analytic conductor: \(3.16207\)
Root analytic conductor: \(1.77822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{396} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 396,\ (\ :1/2),\ 0.918 + 0.396i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25607 - 0.259687i\)
\(L(\frac12)\) \(\approx\) \(1.25607 - 0.259687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 - 1.36i)T \)
3 \( 1 \)
11 \( 1 + T \)
good5 \( 1 + 2.05iT - 5T^{2} \)
7 \( 1 + 4.18iT - 7T^{2} \)
13 \( 1 - 3.27T + 13T^{2} \)
17 \( 1 - 0.332iT - 17T^{2} \)
19 \( 1 + 8.47iT - 19T^{2} \)
23 \( 1 + 3.52T + 23T^{2} \)
29 \( 1 - 9.52iT - 29T^{2} \)
31 \( 1 - 3.95iT - 31T^{2} \)
37 \( 1 + 1.37T + 37T^{2} \)
41 \( 1 + 5.42iT - 41T^{2} \)
43 \( 1 - 1.53iT - 43T^{2} \)
47 \( 1 - 6.93T + 47T^{2} \)
53 \( 1 + 8.83iT - 53T^{2} \)
59 \( 1 + 3.70T + 59T^{2} \)
61 \( 1 - 14.8T + 61T^{2} \)
67 \( 1 + 6.32iT - 67T^{2} \)
71 \( 1 + 5.77T + 71T^{2} \)
73 \( 1 + 8.72T + 73T^{2} \)
79 \( 1 - 13.2iT - 79T^{2} \)
83 \( 1 - 0.984T + 83T^{2} \)
89 \( 1 - 1.46iT - 89T^{2} \)
97 \( 1 - 5.99T + 97T^{2} \)
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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

−11.15790097560729718369442655977, −10.26726716472901759902679923456, −9.038403719142838851045104074052, −8.456384781330463447822761718325, −7.30880442000563854247396679078, −6.69988466345119947933842410845, −5.28838400329463744523290586371, −4.51273456134865882883886316876, −3.50885598590578174310988645262, −0.821147894588993495330444487543, 2.00402129091665141165057498006, 2.97978073604372772667979378369, 4.12137723581635910225304670811, 5.77367236500695092818379237051, 6.04868495890675277940577105990, 7.925093762678555068795812846151, 8.738950789420074186129728018769, 9.803609639227476326457623421102, 10.47963969912118046570300426398, 11.53185583232925167227952263659

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