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Properties

Label 2-3672-3672.3379-c0-0-0
Degree $2$
Conductor $3672$
Sign $0.568 - 0.822i$
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 + (1.63 + 1.14i)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 + (−1.63 + 1.14i)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 + (1.63 + 1.14i)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 + (−1.63 + 1.14i)22-s + (0.766 − 0.642i)24-s + (0.342 − 0.939i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3672\)    =    \(2^{3} \cdot 3^{3} \cdot 17\)
Sign: $0.568 - 0.822i$
Analytic conductor: \(1.83256\)
Root analytic conductor: \(1.35372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3672} (3379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3672,\ (\ :0),\ 0.568 - 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.596464671\)
\(L(\frac12)\) \(\approx\) \(1.596464671\)
\(L(1)\) 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 \( 1 + (-1.63 - 1.14i)T + (0.342 + 0.939i)T^{2} \)
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 \( 1 + (0.484 - 1.03i)T + (-0.642 - 0.766i)T^{2} \)
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 \( 1 + (-0.469 + 1.75i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (1.14 - 1.63i)T + (-0.342 - 0.939i)T^{2} \)
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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

−8.758791170056960746102064763329, −8.139242010767863170553675258742, −7.29155008736154407735052837079, −6.70899408949747060236993104824, −6.32356145086702975320142908318, −4.86050518595256550250940141071, −4.38582987070723776986325902460, −3.54729437131566230349441483994, −2.19146982807247308062756203420, −1.27719691223143802561460939970, 1.19235428332732739454084032831, 2.05214843832620963191905315110, 3.16619826984900730128518587111, 3.68867684585066237366408067727, 4.35770762038828679582096587864, 5.39083844949371439227800200337, 6.61556295439360116974012755571, 7.26172511814984505299862481712, 8.388110991159815492436405405306, 8.613890175088394728245011698836

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