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Properties

Label 2-3645-135.59-c0-0-3
Degree $2$
Conductor $3645$
Sign $0.727 - 0.686i$
Analytic cond. $1.81909$
Root an. cond. $1.34873$
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  + (−1.43 + 1.20i)2-s + (0.439 − 2.49i)4-s + (−0.939 + 0.342i)5-s + (1.43 + 2.49i)8-s + (0.939 − 1.62i)10-s + (−2.70 − 0.984i)16-s + (−0.173 + 0.300i)17-s + (−0.173 − 0.300i)19-s + (0.439 + 2.49i)20-s + (0.266 − 1.50i)23-s + (0.766 − 0.642i)25-s + (−0.326 + 1.85i)31-s + (2.37 − 0.866i)32-s + (−0.113 − 0.642i)34-s + (0.613 + 0.223i)38-s + ⋯
L(s)  = 1  + (−1.43 + 1.20i)2-s + (0.439 − 2.49i)4-s + (−0.939 + 0.342i)5-s + (1.43 + 2.49i)8-s + (0.939 − 1.62i)10-s + (−2.70 − 0.984i)16-s + (−0.173 + 0.300i)17-s + (−0.173 − 0.300i)19-s + (0.439 + 2.49i)20-s + (0.266 − 1.50i)23-s + (0.766 − 0.642i)25-s + (−0.326 + 1.85i)31-s + (2.37 − 0.866i)32-s + (−0.113 − 0.642i)34-s + (0.613 + 0.223i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3645\)    =    \(3^{6} \cdot 5\)
Sign: $0.727 - 0.686i$
Analytic conductor: \(1.81909\)
Root analytic conductor: \(1.34873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3645} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3645,\ (\ :0),\ 0.727 - 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4256726598\)
\(L(\frac12)\) \(\approx\) \(0.4256726598\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.939 - 0.342i)T \)
good2 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.939 - 0.342i)T^{2} \)
11 \( 1 + (-0.766 - 0.642i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
53 \( 1 - 1.53T + T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.766 - 0.642i)T^{2} \)
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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

−8.577347065091471986556174451238, −8.188633236726796233706539989438, −7.35850767033167572505889820119, −6.77122457426919923074650227200, −6.33311922646595650848629177910, −5.21588890274289728065229741595, −4.52451584254735246982064114925, −3.25482152994149404200979986188, −1.97415503454697068282505049397, −0.59216053786114074988228852767, 0.818411285232987218250108680304, 1.89360636568325936514760275043, 2.92490175828837020140944700301, 3.72524283649598164097925075182, 4.36836040833147650588306556614, 5.60522124914372545311094493305, 6.90367342704278733785075911700, 7.60953488240072512641321024474, 7.992594637785512745229878805060, 8.800201893510720247878880929981

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