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Properties

Label 2-3762-209.208-c1-0-48
Degree $2$
Conductor $3762$
Sign $0.356 - 0.934i$
Analytic cond. $30.0397$
Root an. cond. $5.48085$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2.09·5-s + 2.64i·7-s + 8-s + 2.09·10-s + (0.183 + 3.31i)11-s + 5.69·13-s + 2.64i·14-s + 16-s + 6.27i·17-s + (−3.98 − 1.77i)19-s + 2.09·20-s + (0.183 + 3.31i)22-s + 6.58·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.935·5-s + 1.00i·7-s + 0.353·8-s + 0.661·10-s + (0.0552 + 0.998i)11-s + 1.58·13-s + 0.707i·14-s + 0.250·16-s + 1.52i·17-s + (−0.913 − 0.407i)19-s + 0.467·20-s + (0.0391 + 0.706i)22-s + 1.37·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3762\)    =    \(2 \cdot 3^{2} \cdot 11 \cdot 19\)
Sign: $0.356 - 0.934i$
Analytic conductor: \(30.0397\)
Root analytic conductor: \(5.48085\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3762} (2089, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3762,\ (\ :1/2),\ 0.356 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.767382465\)
\(L(\frac12)\) \(\approx\) \(3.767382465\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
11 \( 1 + (-0.183 - 3.31i)T \)
19 \( 1 + (3.98 + 1.77i)T \)
good5 \( 1 - 2.09T + 5T^{2} \)
7 \( 1 - 2.64iT - 7T^{2} \)
13 \( 1 - 5.69T + 13T^{2} \)
17 \( 1 - 6.27iT - 17T^{2} \)
23 \( 1 - 6.58T + 23T^{2} \)
29 \( 1 + 4.30T + 29T^{2} \)
31 \( 1 - 2.68iT - 31T^{2} \)
37 \( 1 - 1.22iT - 37T^{2} \)
41 \( 1 + 9.90T + 41T^{2} \)
43 \( 1 + 9.42iT - 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 + 3.73iT - 53T^{2} \)
59 \( 1 - 0.0279iT - 59T^{2} \)
61 \( 1 - 8.90iT - 61T^{2} \)
67 \( 1 + 15.9iT - 67T^{2} \)
71 \( 1 + 1.33iT - 71T^{2} \)
73 \( 1 - 2.05iT - 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 - 6.89iT - 83T^{2} \)
89 \( 1 - 6.24iT - 89T^{2} \)
97 \( 1 - 9.05iT - 97T^{2} \)
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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

−8.730906898705466758275468389458, −8.005160511604629566839064997142, −6.67508002639865229848256931821, −6.46811093599464403550442361447, −5.58336349210873998850130261679, −5.07886724202649883686105438103, −4.01706262322895071095661483038, −3.23941853728969268645907527189, −2.02790415872803611109030531707, −1.66663747529418024475269957777, 0.842860406477504337689463847613, 1.80551963551099661039233931228, 3.06162045607237118640734693501, 3.62720159660620487911987850256, 4.59131146586347539983566886425, 5.40065405297556161880543167389, 6.15911246684450351973466868272, 6.62540040867755394079449395365, 7.48052206070989617765454869102, 8.365052942977394996046004158458

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