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Properties

Label 2-396-99.43-c0-0-1
Degree $2$
Conductor $396$
Sign $0.766 + 0.642i$
Analytic cond. $0.197629$
Root an. cond. $0.444555$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + (−1 − 1.73i)5-s + 9-s + (−0.5 + 0.866i)11-s + (−1 − 1.73i)15-s + (0.5 + 0.866i)23-s + (−1.49 + 2.59i)25-s + 27-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s − 37-s + (−1 − 1.73i)45-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s − 53-s + ⋯
L(s)  = 1  + 3-s + (−1 − 1.73i)5-s + 9-s + (−0.5 + 0.866i)11-s + (−1 − 1.73i)15-s + (0.5 + 0.866i)23-s + (−1.49 + 2.59i)25-s + 27-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s − 37-s + (−1 − 1.73i)45-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s − 53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(396\)    =    \(2^{2} \cdot 3^{2} \cdot 11\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(0.197629\)
Root analytic conductor: \(0.444555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{396} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 396,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9681479389\)
\(L(\frac12)\) \(\approx\) \(0.9681479389\)
\(L(1)\) 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 \( 1 - T \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−11.70136177925881820698619439201, −10.28648633137896334484492377742, −9.289551231046621611813501617542, −8.662710115179572720583464653053, −7.86856845139893519475827247187, −7.14993318326154384415023470974, −5.19633272524974394269101909565, −4.48479146806682122089604182101, −3.41370750137473105502287963362, −1.60954385196913242968519716196, 2.61582828989042026853766389212, 3.28180609172253435165309953704, 4.33941602508599612843120622822, 6.20915467728156064500234637887, 7.14068872465359179818039593197, 7.87062351680290350812187551406, 8.632733594795348260989589218504, 9.956175888735277583707366660418, 10.71683966024892544806119382673, 11.37822940010574484609253295582

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