Properties

Label 2-72-1.1-c13-0-12
Degree $2$
Conductor $72$
Sign $-1$
Analytic cond. $77.2062$
Root an. cond. $8.78670$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.33e3·5-s − 1.39e5·7-s + 6.48e6·11-s − 2.25e7·13-s + 2.37e7·17-s + 3.25e8·19-s − 9.21e8·23-s − 1.20e9·25-s + 3.86e9·29-s − 2.25e9·31-s − 6.06e8·35-s + 1.82e10·37-s − 3.44e10·41-s − 1.71e10·43-s + 6.73e10·47-s − 7.72e10·49-s + 8.72e10·53-s + 2.80e10·55-s − 5.40e11·59-s − 5.12e10·61-s − 9.78e10·65-s + 2.55e10·67-s + 1.38e12·71-s − 8.19e11·73-s − 9.07e11·77-s − 4.03e12·79-s − 4.18e12·83-s + ⋯
L(s)  = 1  + 0.123·5-s − 0.449·7-s + 1.10·11-s − 1.29·13-s + 0.238·17-s + 1.58·19-s − 1.29·23-s − 0.984·25-s + 1.20·29-s − 0.456·31-s − 0.0557·35-s + 1.16·37-s − 1.13·41-s − 0.414·43-s + 0.911·47-s − 0.797·49-s + 0.540·53-s + 0.136·55-s − 1.66·59-s − 0.127·61-s − 0.160·65-s + 0.0344·67-s + 1.28·71-s − 0.633·73-s − 0.496·77-s − 1.86·79-s − 1.40·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(77.2062\)
Root analytic conductor: \(8.78670\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: $\chi_{72} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 72,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 866 p T + p^{13} T^{2} \)
7 \( 1 + 139992 T + p^{13} T^{2} \)
11 \( 1 - 589484 p T + p^{13} T^{2} \)
13 \( 1 + 22588034 T + p^{13} T^{2} \)
17 \( 1 - 23732270 T + p^{13} T^{2} \)
19 \( 1 - 325344836 T + p^{13} T^{2} \)
23 \( 1 + 921600632 T + p^{13} T^{2} \)
29 \( 1 - 3865879218 T + p^{13} T^{2} \)
31 \( 1 + 2253401440 T + p^{13} T^{2} \)
37 \( 1 - 18250384566 T + p^{13} T^{2} \)
41 \( 1 + 34422845322 T + p^{13} T^{2} \)
43 \( 1 + 17192501444 T + p^{13} T^{2} \)
47 \( 1 - 67371749904 T + p^{13} T^{2} \)
53 \( 1 - 1646815442 p T + p^{13} T^{2} \)
59 \( 1 + 540214518668 T + p^{13} T^{2} \)
61 \( 1 + 51276568850 T + p^{13} T^{2} \)
67 \( 1 - 25519930676 T + p^{13} T^{2} \)
71 \( 1 - 1387500699032 T + p^{13} T^{2} \)
73 \( 1 + 819049441238 T + p^{13} T^{2} \)
79 \( 1 + 4030935615344 T + p^{13} T^{2} \)
83 \( 1 + 4180823831428 T + p^{13} T^{2} \)
89 \( 1 + 2677027798266 T + p^{13} T^{2} \)
97 \( 1 + 14039464316446 T + p^{13} T^{2} \)
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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

−11.68924006563963254491710255514, −10.00141642330623468459929424486, −9.443982449872460039999215152100, −7.907616234211192966946047793005, −6.78434196518475737987691781821, −5.57516325480379431322335132249, −4.17309454727688773817544167271, −2.87318103261419278719767658450, −1.43418488566739748309553425462, 0, 1.43418488566739748309553425462, 2.87318103261419278719767658450, 4.17309454727688773817544167271, 5.57516325480379431322335132249, 6.78434196518475737987691781821, 7.907616234211192966946047793005, 9.443982449872460039999215152100, 10.00141642330623468459929424486, 11.68924006563963254491710255514

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