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Properties

Label 2-396-12.11-c1-0-10
Degree $2$
Conductor $396$
Sign $0.713 - 0.700i$
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  + (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

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

Invariants

Degree: \(2\)
Conductor: \(396\)    =    \(2^{2} \cdot 3^{2} \cdot 11\)
Sign: $0.713 - 0.700i$
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: \((2,\ 396,\ (\ :1/2),\ 0.713 - 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28581 + 0.935108i\)
\(L(\frac12)\) \(\approx\) \(2.28581 + 0.935108i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.126i)T \)
3 \( 1 \)
11 \( 1 - T \)
good5 \( 1 - 0.450iT - 5T^{2} \)
7 \( 1 - 4.60iT - 7T^{2} \)
13 \( 1 + 5.39T + 13T^{2} \)
17 \( 1 + 5.39iT - 17T^{2} \)
19 \( 1 + 4.38iT - 19T^{2} \)
23 \( 1 - 6.03T + 23T^{2} \)
29 \( 1 + 3.31iT - 29T^{2} \)
31 \( 1 - 4.57iT - 31T^{2} \)
37 \( 1 + 6.27T + 37T^{2} \)
41 \( 1 + 11.2iT - 41T^{2} \)
43 \( 1 - 0.650iT - 43T^{2} \)
47 \( 1 + 5.79T + 47T^{2} \)
53 \( 1 + 1.46iT - 53T^{2} \)
59 \( 1 + 4.71T + 59T^{2} \)
61 \( 1 - 1.28T + 61T^{2} \)
67 \( 1 - 6.54iT - 67T^{2} \)
71 \( 1 + 4.27T + 71T^{2} \)
73 \( 1 - 9.86T + 73T^{2} \)
79 \( 1 + 8.84iT - 79T^{2} \)
83 \( 1 - 0.627T + 83T^{2} \)
89 \( 1 - 10.6iT - 89T^{2} \)
97 \( 1 - 1.59T + 97T^{2} \)
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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

−11.78422280892013307963503863133, −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 $Z$-function along the critical line