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Properties

Label 2-3736-3736.147-c0-0-0
Degree $2$
Conductor $3736$
Sign $0.776 + 0.630i$
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  + (−0.311 + 0.950i)2-s + (−0.163 − 0.438i)3-s + (−0.805 − 0.592i)4-s + (0.467 − 0.0189i)6-s + (0.813 − 0.581i)8-s + (0.590 − 0.512i)9-s + (0.332 − 0.399i)11-s + (−0.127 + 0.449i)12-s + (0.298 + 0.954i)16-s + (0.483 − 0.995i)17-s + (0.302 + 0.720i)18-s + (0.00952 − 0.0938i)19-s + (0.275 + 0.440i)22-s + (−0.387 − 0.261i)24-s + (−0.660 − 0.750i)25-s + ⋯
L(s)  = 1  + (−0.311 + 0.950i)2-s + (−0.163 − 0.438i)3-s + (−0.805 − 0.592i)4-s + (0.467 − 0.0189i)6-s + (0.813 − 0.581i)8-s + (0.590 − 0.512i)9-s + (0.332 − 0.399i)11-s + (−0.127 + 0.449i)12-s + (0.298 + 0.954i)16-s + (0.483 − 0.995i)17-s + (0.302 + 0.720i)18-s + (0.00952 − 0.0938i)19-s + (0.275 + 0.440i)22-s + (−0.387 − 0.261i)24-s + (−0.660 − 0.750i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3736\)    =    \(2^{3} \cdot 467\)
Sign: $0.776 + 0.630i$
Analytic conductor: \(1.86450\)
Root analytic conductor: \(1.36546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3736} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3736,\ (\ :0),\ 0.776 + 0.630i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9110905473\)
\(L(\frac12)\) \(\approx\) \(0.9110905473\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.311 - 0.950i)T \)
467 \( 1 + (0.718 + 0.695i)T \)
good3 \( 1 + (0.163 + 0.438i)T + (-0.755 + 0.655i)T^{2} \)
5 \( 1 + (0.660 + 0.750i)T^{2} \)
7 \( 1 + (-0.896 + 0.442i)T^{2} \)
11 \( 1 + (-0.332 + 0.399i)T + (-0.181 - 0.983i)T^{2} \)
13 \( 1 + (-0.986 - 0.161i)T^{2} \)
17 \( 1 + (-0.483 + 0.995i)T + (-0.618 - 0.785i)T^{2} \)
19 \( 1 + (-0.00952 + 0.0938i)T + (-0.979 - 0.200i)T^{2} \)
23 \( 1 + (-0.843 - 0.536i)T^{2} \)
29 \( 1 + (-0.994 + 0.107i)T^{2} \)
31 \( 1 + (0.878 + 0.478i)T^{2} \)
37 \( 1 + (0.999 + 0.0134i)T^{2} \)
41 \( 1 + (0.890 - 1.38i)T + (-0.412 - 0.911i)T^{2} \)
43 \( 1 + (0.988 + 1.09i)T + (-0.100 + 0.994i)T^{2} \)
47 \( 1 + (-0.586 - 0.809i)T^{2} \)
53 \( 1 + (0.973 - 0.227i)T^{2} \)
59 \( 1 + (0.290 - 0.168i)T + (0.496 - 0.868i)T^{2} \)
61 \( 1 + (0.997 + 0.0673i)T^{2} \)
67 \( 1 + (0.980 - 1.61i)T + (-0.460 - 0.887i)T^{2} \)
71 \( 1 + (0.337 + 0.941i)T^{2} \)
73 \( 1 + (-0.362 + 0.00488i)T + (0.999 - 0.0269i)T^{2} \)
79 \( 1 + (-0.929 + 0.368i)T^{2} \)
83 \( 1 + (-0.571 + 1.91i)T + (-0.836 - 0.547i)T^{2} \)
89 \( 1 + (-0.942 + 1.41i)T + (-0.387 - 0.921i)T^{2} \)
97 \( 1 + (-1.33 + 1.44i)T + (-0.0740 - 0.997i)T^{2} \)
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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

−8.576084962342399603457864805497, −7.67772924790211591991789871261, −7.17875461669065321417662734996, −6.45847392759465064098893768579, −5.88934717105492847380853996705, −4.99991574552911742059033195657, −4.19893408782254467004424277647, −3.26067624591651993223989949762, −1.75604863118875958615260433935, −0.64984465180922184744416845309, 1.41393385761618651163796397471, 2.14568958571050156650772348340, 3.44945308137642976107202194080, 3.97502106052986513802055359995, 4.82950467689458078897771743261, 5.50407051172585619896652278319, 6.62400841578247234357546277468, 7.62073429335251402042889947688, 8.049832484831564916868075759782, 9.055993741395606547831239629246

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