Properties

Label 2-72-8.5-c7-0-5
Degree $2$
Conductor $72$
Sign $-0.0484 - 0.998i$
Analytic cond. $22.4917$
Root an. cond. $4.74254$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.70 − 5.81i)2-s + (60.3 − 112. i)4-s + 324. i·5-s − 956.·7-s + (−70.1 − 1.44e3i)8-s + (1.88e3 + 3.14e3i)10-s + 5.45e3i·11-s + 6.28e3i·13-s + (−9.28e3 + 5.56e3i)14-s + (−9.09e3 − 1.36e4i)16-s − 3.45e4·17-s + 1.45e4i·19-s + (3.66e4 + 1.95e4i)20-s + (3.17e4 + 5.29e4i)22-s + 2.46e4·23-s + ⋯
L(s)  = 1  + (0.857 − 0.513i)2-s + (0.471 − 0.881i)4-s + 1.16i·5-s − 1.05·7-s + (−0.0484 − 0.998i)8-s + (0.596 + 0.995i)10-s + 1.23i·11-s + 0.793i·13-s + (−0.904 + 0.541i)14-s + (−0.554 − 0.831i)16-s − 1.70·17-s + 0.488i·19-s + (1.02 + 0.547i)20-s + (0.634 + 1.05i)22-s + 0.422·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0484 - 0.998i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0484 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.0484 - 0.998i$
Analytic conductor: \(22.4917\)
Root analytic conductor: \(4.74254\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :7/2),\ -0.0484 - 0.998i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.15702 + 1.21446i\)
\(L(\frac12)\) \(\approx\) \(1.15702 + 1.21446i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.70 + 5.81i)T \)
3 \( 1 \)
good5 \( 1 - 324. iT - 7.81e4T^{2} \)
7 \( 1 + 956.T + 8.23e5T^{2} \)
11 \( 1 - 5.45e3iT - 1.94e7T^{2} \)
13 \( 1 - 6.28e3iT - 6.27e7T^{2} \)
17 \( 1 + 3.45e4T + 4.10e8T^{2} \)
19 \( 1 - 1.45e4iT - 8.93e8T^{2} \)
23 \( 1 - 2.46e4T + 3.40e9T^{2} \)
29 \( 1 - 1.71e5iT - 1.72e10T^{2} \)
31 \( 1 - 1.11e5T + 2.75e10T^{2} \)
37 \( 1 - 1.03e5iT - 9.49e10T^{2} \)
41 \( 1 + 7.16e4T + 1.94e11T^{2} \)
43 \( 1 - 3.28e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.19e5T + 5.06e11T^{2} \)
53 \( 1 + 1.04e6iT - 1.17e12T^{2} \)
59 \( 1 + 2.25e5iT - 2.48e12T^{2} \)
61 \( 1 + 1.55e6iT - 3.14e12T^{2} \)
67 \( 1 + 3.16e5iT - 6.06e12T^{2} \)
71 \( 1 + 5.38e5T + 9.09e12T^{2} \)
73 \( 1 + 2.68e6T + 1.10e13T^{2} \)
79 \( 1 - 8.22e6T + 1.92e13T^{2} \)
83 \( 1 + 5.89e6iT - 2.71e13T^{2} \)
89 \( 1 + 4.37e5T + 4.42e13T^{2} \)
97 \( 1 + 7.84e6T + 8.07e13T^{2} \)
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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

−13.38954785617075056366465466215, −12.48107137445936482129495947049, −11.29453106634439176504928487295, −10.33139509632270565955506449903, −9.368990626694628900530027430509, −6.91495388914397436267461098013, −6.51850787555933153418523545073, −4.60761908092862012056191107797, −3.24962731139650811800349523121, −2.05307519545334511973140157664, 0.40397294169148143363687917825, 2.82442743394406087709862844087, 4.27014392347832649548468165581, 5.57776960157929975579076387267, 6.63268587082342441404846244904, 8.249835525526135810178408816272, 9.131361520225358944869146114252, 10.94267238325725734511494669486, 12.18331659167327056104803139836, 13.28777633384057606783632631489

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