Properties

Label 2-3724-133.37-c0-0-1
Degree $2$
Conductor $3724$
Sign $0.605 - 0.795i$
Analytic cond. $1.85851$
Root an. cond. $1.36327$
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)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−1 + 1.73i)23-s − 43-s + (0.499 + 0.866i)45-s + (0.5 − 0.866i)47-s + 0.999·55-s + (0.5 − 0.866i)61-s + (0.5 + 0.866i)73-s + (−0.499 − 0.866i)81-s + 2·83-s + 0.999·85-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−1 + 1.73i)23-s − 43-s + (0.499 + 0.866i)45-s + (0.5 − 0.866i)47-s + 0.999·55-s + (0.5 − 0.866i)61-s + (0.5 + 0.866i)73-s + (−0.499 − 0.866i)81-s + 2·83-s + 0.999·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.263252575\)
\(L(\frac12)\) \(\approx\) \(1.263252575\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.717109917227330457579181591885, −8.124982455574306957142741798460, −7.50783802004575073445301186780, −6.48957887246313796201305956739, −5.61881413269342878491488954252, −5.23116276040291331517536910185, −4.23782174999867611170028495022, −3.47791715399893294117677873613, −2.01209835935887637317845673145, −1.56922280159680543184160473022, 0.72593903728398656606338243250, 2.34453309674846738681748061847, 2.97264276023836715606033146379, 3.81565440144836153143760433219, 4.81419628247920123479889106032, 5.85600528650101313618957763664, 6.43768228897130868959087639998, 6.81166801410978967905108111328, 7.909236072443285415643004601586, 8.699629564000920235757333711119

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