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Properties

Label 2-3672-3672.115-c0-0-0
Degree $2$
Conductor $3672$
Sign $0.660 - 0.751i$
Analytic cond. $1.83256$
Root an. cond. $1.35372$
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  + (0.342 + 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 + 0.642i)4-s + (−0.173 − 0.984i)6-s + (−0.866 − 0.500i)8-s + (0.939 + 0.342i)9-s + (−0.0999 − 0.142i)11-s + (0.866 − 0.5i)12-s + (0.173 − 0.984i)16-s + (0.984 − 0.173i)17-s + 0.999i·18-s + (−0.300 − 0.173i)19-s + (0.0999 − 0.142i)22-s + (0.766 + 0.642i)24-s + (−0.342 − 0.939i)25-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 + 0.642i)4-s + (−0.173 − 0.984i)6-s + (−0.866 − 0.500i)8-s + (0.939 + 0.342i)9-s + (−0.0999 − 0.142i)11-s + (0.866 − 0.5i)12-s + (0.173 − 0.984i)16-s + (0.984 − 0.173i)17-s + 0.999i·18-s + (−0.300 − 0.173i)19-s + (0.0999 − 0.142i)22-s + (0.766 + 0.642i)24-s + (−0.342 − 0.939i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3672\)    =    \(2^{3} \cdot 3^{3} \cdot 17\)
Sign: $0.660 - 0.751i$
Analytic conductor: \(1.83256\)
Root analytic conductor: \(1.35372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3672} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3672,\ (\ :0),\ 0.660 - 0.751i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9586665679\)
\(L(\frac12)\) \(\approx\) \(0.9586665679\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 - 0.939i)T \)
3 \( 1 + (0.984 + 0.173i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
good5 \( 1 + (0.342 + 0.939i)T^{2} \)
7 \( 1 + (-0.984 + 0.173i)T^{2} \)
11 \( 1 + (0.0999 + 0.142i)T + (-0.342 + 0.939i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
19 \( 1 + (0.300 + 0.173i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.984 + 0.173i)T^{2} \)
29 \( 1 + (-0.642 - 0.766i)T^{2} \)
31 \( 1 + (-0.984 - 0.173i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (-1.48 + 0.692i)T + (0.642 - 0.766i)T^{2} \)
43 \( 1 + (0.673 + 0.118i)T + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-1.93 + 0.342i)T + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.984 + 0.173i)T^{2} \)
67 \( 1 + (-1.20 - 0.439i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T^{2} \)
73 \( 1 + (0.816 - 0.218i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.642 + 0.766i)T^{2} \)
83 \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \)
89 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.142 + 0.0999i)T + (0.342 - 0.939i)T^{2} \)
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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

−8.523976694290319157378831666618, −7.915133161190276669679435681598, −7.12874366166560132542290051662, −6.60083850828877191803751265984, −5.69883096282738415753559477743, −5.37770939705717428686011640191, −4.39392628612284253774743863300, −3.74449465003731180982451651576, −2.44023845381808497300153847157, −0.77855232300773427426454971306, 0.951913423964843985712042165043, 1.98346985555082650155710949500, 3.22028030164745597288583345698, 4.00983942236847088453913753140, 4.75195037589615163165552542496, 5.57614336773282533802677879436, 5.99427483354326571826396721334, 7.02401908696114244449919824292, 7.87739722364093876482381712318, 8.864305589943386401477814601961

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