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Properties

Label 2-396-11.9-c1-0-0
Degree $2$
Conductor $396$
Sign $0.0219 - 0.999i$
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  + (1.19 + 3.66i)5-s + (0.190 + 0.138i)7-s + (−3.30 − 0.224i)11-s + (−1.92 + 5.93i)13-s + (−0.736 − 2.26i)17-s + (4.11 − 2.99i)19-s + 0.236·23-s + (−7.97 + 5.79i)25-s + (3.61 + 2.62i)29-s + (−1.97 + 6.06i)31-s + (−0.281 + 0.865i)35-s + (3.04 + 2.21i)37-s + (7.66 − 5.56i)41-s + 9.47·43-s + (2.92 − 2.12i)47-s + ⋯
L(s)  = 1  + (0.532 + 1.63i)5-s + (0.0721 + 0.0524i)7-s + (−0.997 − 0.0676i)11-s + (−0.534 + 1.64i)13-s + (−0.178 − 0.549i)17-s + (0.944 − 0.686i)19-s + 0.0492·23-s + (−1.59 + 1.15i)25-s + (0.671 + 0.488i)29-s + (−0.354 + 1.09i)31-s + (−0.0475 + 0.146i)35-s + (0.500 + 0.363i)37-s + (1.19 − 0.869i)41-s + 1.44·43-s + (0.426 − 0.310i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(396\)    =    \(2^{2} \cdot 3^{2} \cdot 11\)
Sign: $0.0219 - 0.999i$
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: \((2,\ 396,\ (\ :1/2),\ 0.0219 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.946065 + 0.925491i\)
\(L(\frac12)\) \(\approx\) \(0.946065 + 0.925491i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.30 + 0.224i)T \)
good5 \( 1 + (-1.19 - 3.66i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.190 - 0.138i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.92 - 5.93i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.736 + 2.26i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.11 + 2.99i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 0.236T + 23T^{2} \)
29 \( 1 + (-3.61 - 2.62i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.97 - 6.06i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.04 - 2.21i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.66 + 5.56i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.47T + 43T^{2} \)
47 \( 1 + (-2.92 + 2.12i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.02 - 6.24i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.73 - 1.26i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.809 + 2.48i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 0.145T + 67T^{2} \)
71 \( 1 + (-0.427 - 1.31i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.61 + 1.90i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.39 + 13.5i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.78 + 14.7i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 + (1.57 - 4.84i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.29521053381075743016219405029, −10.68541031461201324212259317762, −9.786442046067160967132694997287, −8.974257002601614948631534450117, −7.36522311145941299570911195589, −7.01793187072569740014196318802, −5.89682151781803570345903390949, −4.69388884092043938640256424435, −3.10162850736509162765078815712, −2.24583532362425862651233854821, 0.892661504071150381320005476610, 2.57695755750749628541181171258, 4.29634760828092605867898064219, 5.36614886679301631462200702690, 5.83730220338691573113281123472, 7.80741524380577023545258765176, 8.080011065051070081025793445283, 9.370533530028134562307519754992, 9.970885298068377765243593165122, 10.97871546940621577073292177875

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