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Properties

Label 2-3736-3736.51-c0-0-0
Degree $2$
Conductor $3736$
Sign $0.990 - 0.135i$
Analytic cond. $1.86450$
Root an. cond. $1.36546$
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  + (−0.233 + 0.972i)2-s + (0.855 − 0.628i)3-s + (−0.890 − 0.454i)4-s + (0.411 + 0.978i)6-s + (0.650 − 0.759i)8-s + (0.0379 − 0.121i)9-s + (1.87 + 0.544i)11-s + (−1.04 + 0.171i)12-s + (0.586 + 0.809i)16-s + (−0.978 − 1.66i)17-s + (0.108 + 0.0652i)18-s + (0.406 − 1.91i)19-s + (−0.966 + 1.69i)22-s + (0.0786 − 1.05i)24-s + (−0.878 − 0.478i)25-s + ⋯
L(s)  = 1  + (−0.233 + 0.972i)2-s + (0.855 − 0.628i)3-s + (−0.890 − 0.454i)4-s + (0.411 + 0.978i)6-s + (0.650 − 0.759i)8-s + (0.0379 − 0.121i)9-s + (1.87 + 0.544i)11-s + (−1.04 + 0.171i)12-s + (0.586 + 0.809i)16-s + (−0.978 − 1.66i)17-s + (0.108 + 0.0652i)18-s + (0.406 − 1.91i)19-s + (−0.966 + 1.69i)22-s + (0.0786 − 1.05i)24-s + (−0.878 − 0.478i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3736\)    =    \(2^{3} \cdot 467\)
Sign: $0.990 - 0.135i$
Analytic conductor: \(1.86450\)
Root analytic conductor: \(1.36546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3736} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3736,\ (\ :0),\ 0.990 - 0.135i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.509843723\)
\(L(\frac12)\) \(\approx\) \(1.509843723\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.233 - 0.972i)T \)
467 \( 1 + (-0.956 + 0.292i)T \)
good3 \( 1 + (-0.855 + 0.628i)T + (0.298 - 0.954i)T^{2} \)
5 \( 1 + (0.878 + 0.478i)T^{2} \)
7 \( 1 + (-0.746 - 0.665i)T^{2} \)
11 \( 1 + (-1.87 - 0.544i)T + (0.843 + 0.536i)T^{2} \)
13 \( 1 + (0.0202 - 0.999i)T^{2} \)
17 \( 1 + (0.978 + 1.66i)T + (-0.484 + 0.874i)T^{2} \)
19 \( 1 + (-0.406 + 1.91i)T + (-0.913 - 0.405i)T^{2} \)
23 \( 1 + (0.755 - 0.655i)T^{2} \)
29 \( 1 + (0.999 - 0.0134i)T^{2} \)
31 \( 1 + (-0.324 - 0.945i)T^{2} \)
37 \( 1 + (0.924 - 0.381i)T^{2} \)
41 \( 1 + (-0.492 - 0.382i)T + (0.246 + 0.969i)T^{2} \)
43 \( 1 + (-1.11 - 1.37i)T + (-0.207 + 0.978i)T^{2} \)
47 \( 1 + (0.618 + 0.785i)T^{2} \)
53 \( 1 + (0.934 + 0.356i)T^{2} \)
59 \( 1 + (1.76 - 0.871i)T + (0.608 - 0.793i)T^{2} \)
61 \( 1 + (-0.374 - 0.927i)T^{2} \)
67 \( 1 + (0.239 + 1.86i)T + (-0.967 + 0.253i)T^{2} \)
71 \( 1 + (-0.519 + 0.854i)T^{2} \)
73 \( 1 + (1.56 + 0.643i)T + (0.709 + 0.704i)T^{2} \)
79 \( 1 + (0.0471 + 0.998i)T^{2} \)
83 \( 1 + (-1.57 + 0.970i)T + (0.448 - 0.893i)T^{2} \)
89 \( 1 + (-1.82 - 0.505i)T + (0.858 + 0.513i)T^{2} \)
97 \( 1 + (-0.207 + 0.694i)T + (-0.836 - 0.547i)T^{2} \)
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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

−8.875865268717912581736005325267, −7.65525627362668565282383161698, −7.40517331681695862143291913083, −6.66398909582333280225023142685, −6.11271114210054092506414470993, −4.71501280844701297547716316448, −4.49374515280498283302446044568, −3.16854537497690005009378981949, −2.17473728393970834143744494050, −0.993463269598820899933127394475, 1.36132412808467508206721756331, 2.15971892492083464812107442924, 3.46997674278434354719700544366, 3.86001816730836891088844031078, 4.21480370126347960076096856409, 5.70576887616066272310311107288, 6.31056205995193845775612534144, 7.55010700854017670372062262412, 8.357863374770511276299906950405, 8.874216292202150052544138499635

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