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Properties

Label 2-396-11.9-c1-0-3
Degree $2$
Conductor $396$
Sign $0.263 + 0.964i$
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  + (−0.190 − 0.587i)5-s + (−2.30 − 1.67i)7-s + (3.23 − 0.726i)11-s + (1.42 − 4.39i)13-s + (−1.42 − 4.39i)17-s + (2.30 − 1.67i)19-s − 6.47·23-s + (3.73 − 2.71i)25-s + (5.16 + 3.75i)29-s + (−1.80 + 5.56i)31-s + (−0.545 + 1.67i)35-s + (−3.92 − 2.85i)37-s + (5.16 − 3.75i)41-s + (2.92 − 2.12i)47-s + (0.354 + 1.08i)49-s + ⋯
L(s)  = 1  + (−0.0854 − 0.262i)5-s + (−0.872 − 0.634i)7-s + (0.975 − 0.219i)11-s + (0.395 − 1.21i)13-s + (−0.346 − 1.06i)17-s + (0.529 − 0.384i)19-s − 1.34·23-s + (0.747 − 0.542i)25-s + (0.958 + 0.696i)29-s + (−0.324 + 0.999i)31-s + (−0.0921 + 0.283i)35-s + (−0.645 − 0.469i)37-s + (0.806 − 0.585i)41-s + (0.426 − 0.310i)47-s + (0.0505 + 0.155i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(396\)    =    \(2^{2} \cdot 3^{2} \cdot 11\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(3.16207\)
Root analytic conductor: \(1.77822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{396} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 396,\ (\ :1/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.936745 - 0.715215i\)
\(L(\frac12)\) \(\approx\) \(0.936745 - 0.715215i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (-3.23 + 0.726i)T \)
good5 \( 1 + (0.190 + 0.587i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.30 + 1.67i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.42 + 4.39i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.42 + 4.39i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.30 + 1.67i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 + (-5.16 - 3.75i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.80 - 5.56i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.92 + 2.85i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.16 + 3.75i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-2.92 + 2.12i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.19 - 6.74i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.16 + 5.93i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.42 - 4.39i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 4.94T + 67T^{2} \)
71 \( 1 + (-2.66 - 8.19i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-9.78 - 7.10i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.28 - 13.1i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.95 - 15.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 8.47T + 89T^{2} \)
97 \( 1 + (-1.71 + 5.29i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.99417444892206014286435774181, −10.20903352663760162406501097539, −9.289469587024546057478928894591, −8.431931385448368413251499561505, −7.23201979188717192383296450109, −6.45703604990058265734393677413, −5.27926744391565641597606008954, −3.98733671883468574370914844850, −2.96814174865763253975950217129, −0.816348048498922925798932086761, 1.88735767451518521078901349659, 3.43363927833340893459878816808, 4.40863786205955297066007152770, 6.15560255631355031775978571589, 6.42474598914076281619693840328, 7.77156249221346300027204044322, 8.950218425281242666285468341482, 9.513338313352230665340927047368, 10.53446208597995971564489562773, 11.67459643263741548990241090125

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