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Properties

Label 2-396-44.43-c1-0-6
Degree $2$
Conductor $396$
Sign $0.978 - 0.207i$
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  + (−0.430 − 1.34i)2-s + (−1.62 + 1.15i)4-s − 2.93·5-s − 0.348·7-s + (2.26 + 1.69i)8-s + (1.26 + 3.95i)10-s + (3.04 + 1.31i)11-s + 1.83i·13-s + (0.150 + 0.469i)14-s + (1.30 − 3.77i)16-s + 3.20i·17-s + 7.57·19-s + (4.78 − 3.40i)20-s + (0.467 − 4.66i)22-s + 2.93i·23-s + ⋯
L(s)  = 1  + (−0.304 − 0.952i)2-s + (−0.814 + 0.579i)4-s − 1.31·5-s − 0.131·7-s + (0.800 + 0.599i)8-s + (0.400 + 1.25i)10-s + (0.917 + 0.397i)11-s + 0.508i·13-s + (0.0401 + 0.125i)14-s + (0.327 − 0.944i)16-s + 0.777i·17-s + 1.73·19-s + (1.07 − 0.762i)20-s + (0.0997 − 0.995i)22-s + 0.612i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(396\)    =    \(2^{2} \cdot 3^{2} \cdot 11\)
Sign: $0.978 - 0.207i$
Analytic conductor: \(3.16207\)
Root analytic conductor: \(1.77822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{396} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 396,\ (\ :1/2),\ 0.978 - 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.765038 + 0.0803967i\)
\(L(\frac12)\) \(\approx\) \(0.765038 + 0.0803967i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.430 + 1.34i)T \)
3 \( 1 \)
11 \( 1 + (-3.04 - 1.31i)T \)
good5 \( 1 + 2.93T + 5T^{2} \)
7 \( 1 + 0.348T + 7T^{2} \)
13 \( 1 - 1.83iT - 13T^{2} \)
17 \( 1 - 3.20iT - 17T^{2} \)
19 \( 1 - 7.57T + 19T^{2} \)
23 \( 1 - 2.93iT - 23T^{2} \)
29 \( 1 - 5.84iT - 29T^{2} \)
31 \( 1 - 6.51iT - 31T^{2} \)
37 \( 1 + 4.63T + 37T^{2} \)
41 \( 1 + 9.98iT - 41T^{2} \)
43 \( 1 - 0.237T + 43T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 - 8.21T + 53T^{2} \)
59 \( 1 - 1.36iT - 59T^{2} \)
61 \( 1 + 5.97iT - 61T^{2} \)
67 \( 1 + 0.639iT - 67T^{2} \)
71 \( 1 + 9.57iT - 71T^{2} \)
73 \( 1 - 14.9iT - 73T^{2} \)
79 \( 1 + 4.01T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + 9.15T + 89T^{2} \)
97 \( 1 - 1.87T + 97T^{2} \)
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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

−11.44059559379417948517556224699, −10.58151265371946808670306903257, −9.475943235763298632846809719781, −8.769276673389161968084476051264, −7.72774431630676119343440674182, −6.95780317050140761118509340706, −5.14061929854940575958564895251, −3.96872010430445929924835497834, −3.31626529628205309342710414705, −1.40651286137687673815646600590, 0.65430111562848531141642591195, 3.40094694952924317511467414554, 4.42033107367787448000593160024, 5.57962973598368976796880887552, 6.74304599721106947911566171067, 7.59327415960027241678951169841, 8.256008498716969384129230098757, 9.267954232736995696436263954971, 10.09678999419815520185231420740, 11.45851305606279640768514656268

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