Properties

Label 2-72-8.3-c14-0-42
Degree $2$
Conductor $72$
Sign $1$
Analytic cond. $89.5168$
Root an. cond. $9.46133$
Motivic weight $14$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 128·2-s + 1.63e4·4-s + 2.09e6·8-s − 3.87e7·11-s + 2.68e8·16-s + 3.28e8·17-s + 1.77e9·19-s − 4.95e9·22-s + 6.10e9·25-s + 3.43e10·32-s + 4.20e10·34-s + 2.27e11·38-s + 3.33e11·41-s − 4.95e11·43-s − 6.34e11·44-s + 6.78e11·49-s + 7.81e11·50-s + 3.91e12·59-s + 4.39e12·64-s + 2.71e12·67-s + 5.38e12·68-s + 1.57e12·73-s + 2.91e13·76-s + 4.26e13·82-s + 3.11e13·83-s − 6.33e13·86-s − 8.11e13·88-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s − 1.98·11-s + 16-s + 0.800·17-s + 1.99·19-s − 1.98·22-s + 25-s + 32-s + 0.800·34-s + 1.99·38-s + 1.71·41-s − 1.82·43-s − 1.98·44-s + 49-s + 50-s + 1.57·59-s + 64-s + 0.447·67-s + 0.800·68-s + 0.142·73-s + 1.99·76-s + 1.71·82-s + 1.14·83-s − 1.82·86-s − 1.98·88-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(89.5168\)
Root analytic conductor: \(9.46133\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: $\chi_{72} (19, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :7),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(4.516713399\)
\(L(\frac12)\) \(\approx\) \(4.516713399\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{7} T \)
3 \( 1 \)
good5 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
7 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
11 \( 1 + 38712254 T + p^{14} T^{2} \)
13 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
17 \( 1 - 328636222 T + p^{14} T^{2} \)
19 \( 1 - 1778973806 T + p^{14} T^{2} \)
23 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
29 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
31 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
37 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
41 \( 1 - 333393570766 T + p^{14} T^{2} \)
43 \( 1 + 495012562114 T + p^{14} T^{2} \)
47 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
53 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
59 \( 1 - 3914494552162 T + p^{14} T^{2} \)
61 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
67 \( 1 - 2711103884558 T + p^{14} T^{2} \)
71 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
73 \( 1 - 1579402558802 T + p^{14} T^{2} \)
79 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
83 \( 1 - 31146255762898 T + p^{14} T^{2} \)
89 \( 1 - 38433671549134 T + p^{14} T^{2} \)
97 \( 1 + 62815034524126 T + p^{14} T^{2} \)
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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

−11.97833932467342498714599457089, −10.84114136326638267150133816611, −9.900240910353117319822919512147, −8.013885511264961641381760071637, −7.18973849136480687946198306079, −5.60017094213691640254150466637, −4.98694069139826053690994285964, −3.36266398789216184001763274609, −2.49911322371560099801870275950, −0.923254597378679866398334250530, 0.923254597378679866398334250530, 2.49911322371560099801870275950, 3.36266398789216184001763274609, 4.98694069139826053690994285964, 5.60017094213691640254150466637, 7.18973849136480687946198306079, 8.013885511264961641381760071637, 9.900240910353117319822919512147, 10.84114136326638267150133816611, 11.97833932467342498714599457089

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