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Properties

Label 2-3696-924.527-c0-0-2
Degree $2$
Conductor $3696$
Sign $0.553 + 0.832i$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
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  + (−0.5 − 0.866i)3-s − i·7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.866 + 1.5i)17-s + (0.866 − 1.5i)19-s + (−0.866 + 0.5i)21-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 0.999·27-s + 1.73·29-s + (0.499 − 0.866i)33-s + (−0.5 + 0.866i)37-s + 1.73·43-s + (0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s − i·7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.866 + 1.5i)17-s + (0.866 − 1.5i)19-s + (−0.866 + 0.5i)21-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 0.999·27-s + 1.73·29-s + (0.499 − 0.866i)33-s + (−0.5 + 0.866i)37-s + 1.73·43-s + (0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3696\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3696} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3696,\ (\ :0),\ 0.553 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.139385515\)
\(L(\frac12)\) \(\approx\) \(1.139385515\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + iT \)
11 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.73T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.73T + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.316223530501859399112130561667, −7.74083073484682719943717750937, −7.07383226351345442075283378580, −6.52004389492157449349924192244, −5.73960657382913633448098603486, −4.78895785868664511493085704001, −4.08739614111026421015108627442, −2.99177679501891293444263754764, −1.80316370991044765860056831780, −0.939801620321831175139761954884, 1.05506686823130912823402375973, 2.68530462217511191840470005439, 3.35433299179102126857536447240, 4.25372941532370120492249155940, 5.21631799692577963151503243230, 5.77077498139641533673614096186, 6.24549845137279176924385523724, 7.39829174920337093904037571330, 8.208646365163624964455151833822, 9.047327653474492354652292552353

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