Properties

Label 2-3645-135.29-c0-0-6
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  + (0.0603 − 0.342i)2-s + (0.826 + 0.300i)4-s + (0.766 − 0.642i)5-s + (0.326 − 0.565i)8-s + (−0.173 − 0.300i)10-s + (0.500 + 0.419i)16-s + (−0.766 − 1.32i)17-s + (−0.766 + 1.32i)19-s + (0.826 − 0.300i)20-s + (1.76 + 0.642i)23-s + (0.173 − 0.984i)25-s + (−0.326 − 0.118i)31-s + (0.673 − 0.565i)32-s + (−0.5 + 0.181i)34-s + (0.407 + 0.342i)38-s + ⋯
L(s)  = 1  + (0.0603 − 0.342i)2-s + (0.826 + 0.300i)4-s + (0.766 − 0.642i)5-s + (0.326 − 0.565i)8-s + (−0.173 − 0.300i)10-s + (0.500 + 0.419i)16-s + (−0.766 − 1.32i)17-s + (−0.766 + 1.32i)19-s + (0.826 − 0.300i)20-s + (1.76 + 0.642i)23-s + (0.173 − 0.984i)25-s + (−0.326 − 0.118i)31-s + (0.673 − 0.565i)32-s + (−0.5 + 0.181i)34-s + (0.407 + 0.342i)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: $\chi_{3645} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3645,\ (\ :0),\ 0.727 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.921574591\)
\(L(\frac12)\) \(\approx\) \(1.921574591\)
\(L(1)\) 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 \( 1 + (-0.766 + 0.642i)T \)
good2 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.766 + 0.642i)T^{2} \)
11 \( 1 + (-0.173 - 0.984i)T^{2} \)
13 \( 1 + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
53 \( 1 + 1.87T + T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.173 - 0.984i)T^{2} \)
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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

−8.758283950682176887616353441787, −7.85579237302073542476459743206, −7.09271564922238335531188069592, −6.45448931411742253017259422914, −5.62489656568146052383363076377, −4.86778774918380191057496749405, −3.92271035181793859037287558594, −2.91494474811741007367366174429, −2.13360677738802826792739211315, −1.21824039493740852491750852536, 1.47466988725757144303849577943, 2.41350302217785161405107170727, 3.02029793963083969018237336923, 4.35850149649401857880614511282, 5.20987398475069636091032738990, 6.03717294030735812563930936575, 6.64777082155250215483778588098, 6.99740031936594991179140324554, 7.945705027002808777816694985361, 8.879774838224740864699612670338

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