Properties

Label 2-3645-135.104-c0-0-5
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  + (1.76 + 0.642i)2-s + (1.93 + 1.62i)4-s + (0.173 − 0.984i)5-s + (1.43 + 2.49i)8-s + (0.939 − 1.62i)10-s + (0.500 + 2.83i)16-s + (−0.173 + 0.300i)17-s + (−0.173 − 0.300i)19-s + (1.93 − 1.62i)20-s + (1.17 + 0.984i)23-s + (−0.939 − 0.342i)25-s + (−1.43 − 1.20i)31-s + (−0.439 + 2.49i)32-s + (−0.5 + 0.419i)34-s + (−0.113 − 0.642i)38-s + ⋯
L(s)  = 1  + (1.76 + 0.642i)2-s + (1.93 + 1.62i)4-s + (0.173 − 0.984i)5-s + (1.43 + 2.49i)8-s + (0.939 − 1.62i)10-s + (0.500 + 2.83i)16-s + (−0.173 + 0.300i)17-s + (−0.173 − 0.300i)19-s + (1.93 − 1.62i)20-s + (1.17 + 0.984i)23-s + (−0.939 − 0.342i)25-s + (−1.43 − 1.20i)31-s + (−0.439 + 2.49i)32-s + (−0.5 + 0.419i)34-s + (−0.113 − 0.642i)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: $\chi_{3645} (1619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3645,\ (\ :0),\ 0.727 - 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.807648008\)
\(L(\frac12)\) \(\approx\) \(3.807648008\)
\(L(1)\) 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 \( 1 + (-0.173 + 0.984i)T \)
good2 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (-0.173 + 0.984i)T^{2} \)
11 \( 1 + (0.939 - 0.342i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
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 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
53 \( 1 - 1.53T + T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.939 - 0.342i)T^{2} \)
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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

−8.667890404733634583750554554184, −7.69889265894080952352959695938, −7.23065583508529052611324540568, −6.30061079503674580295732323450, −5.59987141698289449157098568805, −5.13049147286634797003275121293, −4.31208156570756221354274191830, −3.71034563220738140796040461045, −2.69140279608575587313450597364, −1.64367836997161302854682438377, 1.57202362486051156431983434491, 2.52790400229205340971772229740, 3.15887600556667301648368501918, 3.88686725099966458069877132586, 4.78271402592914418445032846876, 5.46864233032470297972382636953, 6.24072072331768491330901467184, 6.88126312677216152728999104212, 7.38942405098902934723857911755, 8.699052242379473841232192749300

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