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Properties

Label 2-72-8.5-c3-0-8
Degree $2$
Conductor $72$
Sign $-0.207 + 0.978i$
Analytic cond. $4.24813$
Root an. cond. $2.06110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.34 − 1.58i)2-s + (2.99 + 7.41i)4-s − 6.32i·5-s + 10·7-s + (4.69 − 22.1i)8-s + (−10.0 + 14.8i)10-s − 37.9i·11-s − 59.3i·13-s + (−23.4 − 15.8i)14-s + (−46.0 + 44.4i)16-s − 75.0·17-s − 118. i·19-s + (46.9 − 18.9i)20-s + (−60.0 + 88.9i)22-s + 150.·23-s + ⋯
L(s)  = 1  + (−0.829 − 0.559i)2-s + (0.374 + 0.927i)4-s − 0.565i·5-s + 0.539·7-s + (0.207 − 0.978i)8-s + (−0.316 + 0.469i)10-s − 1.04i·11-s − 1.26i·13-s + (−0.447 − 0.301i)14-s + (−0.718 + 0.695i)16-s − 1.07·17-s − 1.43i·19-s + (0.524 − 0.212i)20-s + (−0.581 + 0.862i)22-s + 1.36·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.207 + 0.978i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ -0.207 + 0.978i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.592975 - 0.731787i\)
\(L(\frac12)\) \(\approx\) \(0.592975 - 0.731787i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.34 + 1.58i)T \)
3 \( 1 \)
good5 \( 1 + 6.32iT - 125T^{2} \)
7 \( 1 - 10T + 343T^{2} \)
11 \( 1 + 37.9iT - 1.33e3T^{2} \)
13 \( 1 + 59.3iT - 2.19e3T^{2} \)
17 \( 1 + 75.0T + 4.91e3T^{2} \)
19 \( 1 + 118. iT - 6.85e3T^{2} \)
23 \( 1 - 150.T + 1.21e4T^{2} \)
29 \( 1 - 246. iT - 2.43e4T^{2} \)
31 \( 1 - 62T + 2.97e4T^{2} \)
37 \( 1 + 59.3iT - 5.06e4T^{2} \)
41 \( 1 + 375.T + 6.89e4T^{2} \)
43 \( 1 - 118. iT - 7.95e4T^{2} \)
47 \( 1 - 450.T + 1.03e5T^{2} \)
53 \( 1 - 132. iT - 1.48e5T^{2} \)
59 \( 1 + 733. iT - 2.05e5T^{2} \)
61 \( 1 - 533. iT - 2.26e5T^{2} \)
67 \( 1 - 711. iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 30T + 3.89e5T^{2} \)
79 \( 1 - 94T + 4.93e5T^{2} \)
83 \( 1 - 670. iT - 5.71e5T^{2} \)
89 \( 1 - 750.T + 7.04e5T^{2} \)
97 \( 1 - 130T + 9.12e5T^{2} \)
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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

−13.43166472160846117522934532718, −12.69159344472675627151117507668, −11.26810327354987394760084865014, −10.68271395759717941866855520390, −9.006553114226092258440059070768, −8.437182450692687677778447787886, −6.96759718286122523807758828622, −4.98240964575716959785315717052, −2.98964440745171456406232368856, −0.843581455303045577154825550162, 1.95698788669787030287002228668, 4.66818566401842876303810128580, 6.43642249899004322837223289722, 7.36016460731941364759068163279, 8.683527573471398514215268319860, 9.829792816275799073305332603816, 10.91081478722974632289438432422, 11.93895753174482768947125277454, 13.72706490407965680655699960224, 14.75626615954404905656443483582

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