Properties

Label 2-3645-135.119-c0-0-2
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  + (1.17 + 0.984i)2-s + (0.233 + 1.32i)4-s + (−0.939 − 0.342i)5-s + (−0.266 + 0.460i)8-s + (−0.766 − 1.32i)10-s + (0.500 − 0.181i)16-s + (0.939 + 1.62i)17-s + (0.939 − 1.62i)19-s + (0.233 − 1.32i)20-s + (0.0603 + 0.342i)23-s + (0.766 + 0.642i)25-s + (0.266 + 1.50i)31-s + (1.26 + 0.460i)32-s + (−0.5 + 2.83i)34-s + (2.70 − 0.984i)38-s + ⋯
L(s)  = 1  + (1.17 + 0.984i)2-s + (0.233 + 1.32i)4-s + (−0.939 − 0.342i)5-s + (−0.266 + 0.460i)8-s + (−0.766 − 1.32i)10-s + (0.500 − 0.181i)16-s + (0.939 + 1.62i)17-s + (0.939 − 1.62i)19-s + (0.233 − 1.32i)20-s + (0.0603 + 0.342i)23-s + (0.766 + 0.642i)25-s + (0.266 + 1.50i)31-s + (1.26 + 0.460i)32-s + (−0.5 + 2.83i)34-s + (2.70 − 0.984i)38-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: $\chi_{3645} (3239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3645,\ (\ :0),\ 0.230 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.190091225\)
\(L(\frac12)\) \(\approx\) \(2.190091225\)
\(L(1)\) 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 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
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 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
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 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
53 \( 1 - 0.347T + T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
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 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
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

−8.484536189988351654993678577850, −7.953484413163026698990027443283, −7.23177730983164628004134675075, −6.67026553578032867712754098574, −5.78109404536826753041124609350, −5.04242280866089075480025011482, −4.51909187033611205591406762157, −3.57157938072373756886995600724, −3.10076385379067325136598792352, −1.26573726465828291764736048877, 1.08121514976105724498504027648, 2.42458761088607019296065043260, 3.18597872703368804950825892926, 3.77468251258003511151471193308, 4.54175293233154087929355376088, 5.34334833422663929517485494211, 5.99471296437535775448375952620, 7.14755724686137351307414129941, 7.73804286406415329318442514398, 8.416894667609464000451419350625

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