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Properties

Label 2-3645-135.14-c0-0-0
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  + (−0.266 − 1.50i)2-s + (−1.26 + 0.460i)4-s + (−0.766 − 0.642i)5-s + (0.266 + 0.460i)8-s + (−0.766 + 1.32i)10-s + (−0.407 + 0.342i)16-s + (−0.939 + 1.62i)17-s + (0.939 + 1.62i)19-s + (1.26 + 0.460i)20-s + (0.326 − 0.118i)23-s + (0.173 + 0.984i)25-s + (−1.43 + 0.524i)31-s + (1.03 + 0.866i)32-s + (2.70 + 0.984i)34-s + (2.20 − 1.85i)38-s + ⋯
L(s)  = 1  + (−0.266 − 1.50i)2-s + (−1.26 + 0.460i)4-s + (−0.766 − 0.642i)5-s + (0.266 + 0.460i)8-s + (−0.766 + 1.32i)10-s + (−0.407 + 0.342i)16-s + (−0.939 + 1.62i)17-s + (0.939 + 1.62i)19-s + (1.26 + 0.460i)20-s + (0.326 − 0.118i)23-s + (0.173 + 0.984i)25-s + (−1.43 + 0.524i)31-s + (1.03 + 0.866i)32-s + (2.70 + 0.984i)34-s + (2.20 − 1.85i)38-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: $\chi_{3645} (2834, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3645,\ (\ :0),\ 0.957 + 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5279072077\)
\(L(\frac12)\) \(\approx\) \(0.5279072077\)
\(L(1)\) 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 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
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 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
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 \( 1 + 0.347T + T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
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 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
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

−8.814748185280411468423177075827, −8.244651075216763627430717990307, −7.50257302724548445337270521014, −6.43085355174188743833996786674, −5.48434789997848187884558554269, −4.49345806804556776517615087665, −3.77901751627453704275834267805, −3.29709405123511897189571428868, −1.94944584391619836303582294771, −1.27863513969111598989428988487, 0.35563376167135939988379420198, 2.48127708632949201211611388845, 3.30017500722899932065874918617, 4.58581335399057711969887590909, 5.00204055613944564378515554890, 5.99989380997831250081246181092, 6.81523832302034333083246950160, 7.34062680921395021789446877603, 7.57157848662839008671540637236, 8.724786707899090744216956524661

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