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Properties

Label 2-3645-135.104-c0-0-0
Degree $2$
Conductor $3645$
Sign $-0.230 - 0.973i$
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  + (−0.939 − 0.342i)2-s + (−0.173 + 0.984i)5-s + (0.499 + 0.866i)8-s + (0.5 − 0.866i)10-s + (−0.173 − 0.984i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.766 − 0.642i)31-s + (0.766 − 0.642i)34-s + (−0.173 − 0.984i)38-s + (−0.939 + 0.342i)40-s + (−0.5 − 0.866i)46-s + (−1.53 + 1.28i)47-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.173 + 0.984i)5-s + (0.499 + 0.866i)8-s + (0.5 − 0.866i)10-s + (−0.173 − 0.984i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.766 − 0.642i)31-s + (0.766 − 0.642i)34-s + (−0.173 − 0.984i)38-s + (−0.939 + 0.342i)40-s + (−0.5 − 0.866i)46-s + (−1.53 + 1.28i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3645\)    =    \(3^{6} \cdot 5\)
Sign: $-0.230 - 0.973i$
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: \((2,\ 3645,\ (\ :0),\ -0.230 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4656311250\)
\(L(\frac12)\) \(\approx\) \(0.4656311250\)
\(L(1)\) 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 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
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 \( 1 + (1.53 - 1.28i)T + (0.173 - 0.984i)T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
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 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
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

−9.049314210911441722342520964227, −8.173306778590585365621316716215, −7.68126884148236593537892841860, −6.90428425743757583774987484919, −6.01744093592913221152294665104, −5.27870194288078820679295773745, −4.17674372787268232015863668044, −3.33987648672607902671165217875, −2.30777041134171095976415363702, −1.40083980596807246969160995650, 0.40325269430920316962504812392, 1.50762449235076007186700718727, 2.90350869716948366315388397841, 3.99039677008518975206402101424, 4.78730389520930887368923593221, 5.35939091816205920623952527352, 6.66360930654074091325607898760, 7.13232667443589014139675014175, 7.958972809043822103653301670653, 8.558008821543746013523204286970

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