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Properties

Label 2-3645-135.59-c0-0-1
Degree $2$
Conductor $3645$
Sign $-0.957 - 0.286i$
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  + (−0.766 + 0.642i)2-s + (−0.939 + 0.342i)5-s + (−0.499 − 0.866i)8-s + (0.5 − 0.866i)10-s + (0.939 + 0.342i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.173 + 0.984i)31-s + (0.173 + 0.984i)34-s + (−0.939 − 0.342i)38-s + (0.766 + 0.642i)40-s + (−0.5 − 0.866i)46-s + (0.347 + 1.96i)47-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (−0.939 + 0.342i)5-s + (−0.499 − 0.866i)8-s + (0.5 − 0.866i)10-s + (0.939 + 0.342i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.173 + 0.984i)31-s + (0.173 + 0.984i)34-s + (−0.939 − 0.342i)38-s + (0.766 + 0.642i)40-s + (−0.5 − 0.866i)46-s + (0.347 + 1.96i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3645\)    =    \(3^{6} \cdot 5\)
Sign: $-0.957 - 0.286i$
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: \((2,\ 3645,\ (\ :0),\ -0.957 - 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4525934657\)
\(L(\frac12)\) \(\approx\) \(0.4525934657\)
\(L(1)\) 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 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
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 \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
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 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
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

−8.978845207606328541271787010768, −7.929964727361603878114096638102, −7.83633548712610345262543273185, −7.07243496524060882261879856842, −6.38145692455421572384444005106, −5.44873230197759633240136283813, −4.43340708462142710077882539718, −3.48699743339343711447598302716, −2.98791396666301419606232888925, −1.23515980323002493233666220642, 0.39039799677121155815876909475, 1.54330409116017070381441533477, 2.64299498112681813761917726903, 3.57099887046083271141538820372, 4.50076985377928656162454540485, 5.29465085499758873120679671052, 6.14451129927002733064659367475, 7.12881904652342990543213036543, 7.924267155224486419945892409552, 8.520796087162132686261864995829

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