Properties

Label 2-72-8.3-c12-0-24
Degree $2$
Conductor $72$
Sign $1$
Analytic cond. $65.8075$
Root an. cond. $8.11218$
Motivic weight $12$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 64·2-s + 4.09e3·4-s − 2.62e5·8-s − 1.92e6·11-s + 1.67e7·16-s + 4.52e7·17-s − 8.79e7·19-s + 1.23e8·22-s + 2.44e8·25-s − 1.07e9·32-s − 2.89e9·34-s + 5.62e9·38-s − 8.62e9·41-s − 7.03e9·43-s − 7.87e9·44-s + 1.38e10·49-s − 1.56e10·50-s − 8.63e9·59-s + 6.87e10·64-s + 1.75e11·67-s + 1.85e11·68-s + 4.91e10·73-s − 3.60e11·76-s + 5.52e11·82-s + 1.92e11·83-s + 4.49e11·86-s + 5.04e11·88-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s − 1.08·11-s + 16-s + 1.87·17-s − 1.86·19-s + 1.08·22-s + 25-s − 32-s − 1.87·34-s + 1.86·38-s − 1.81·41-s − 1.11·43-s − 1.08·44-s + 49-s − 50-s − 0.204·59-s + 64-s + 1.93·67-s + 1.87·68-s + 0.324·73-s − 1.86·76-s + 1.81·82-s + 0.590·83-s + 1.11·86-s + 1.08·88-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(65.8075\)
Root analytic conductor: \(8.11218\)
Motivic weight: \(12\)
Rational: yes
Arithmetic: yes
Character: $\chi_{72} (19, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :6),\ 1)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.032907053\)
\(L(\frac12)\) \(\approx\) \(1.032907053\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{6} T \)
3 \( 1 \)
good5 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
7 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
11 \( 1 + 1923122 T + p^{12} T^{2} \)
13 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
17 \( 1 - 45296062 T + p^{12} T^{2} \)
19 \( 1 + 87931438 T + p^{12} T^{2} \)
23 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
29 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
31 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
37 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
41 \( 1 + 8628259682 T + p^{12} T^{2} \)
43 \( 1 + 7030618702 T + p^{12} T^{2} \)
47 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
53 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
59 \( 1 + 8638314482 T + p^{12} T^{2} \)
61 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
67 \( 1 - 175045819538 T + p^{12} T^{2} \)
71 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
73 \( 1 - 49139489378 T + p^{12} T^{2} \)
79 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
83 \( 1 - 192940233262 T + p^{12} T^{2} \)
89 \( 1 - 866326445278 T + p^{12} T^{2} \)
97 \( 1 - 1656488134658 T + p^{12} T^{2} \)
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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

−12.01327197133192754076936257447, −10.66782154489306705865091336129, −10.06674718532524411932992872016, −8.643758172068792181719992313412, −7.81596081253419646631743599396, −6.57398016214277117871036339438, −5.25394708590742550536800017901, −3.28164968065611900867587269377, −2.01842683137815890590778162772, −0.61014332829218763801154699599, 0.61014332829218763801154699599, 2.01842683137815890590778162772, 3.28164968065611900867587269377, 5.25394708590742550536800017901, 6.57398016214277117871036339438, 7.81596081253419646631743599396, 8.643758172068792181719992313412, 10.06674718532524411932992872016, 10.66782154489306705865091336129, 12.01327197133192754076936257447

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