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Properties

Label 2-72-72.29-c2-0-18
Degree $2$
Conductor $72$
Sign $0.687 + 0.726i$
Analytic cond. $1.96185$
Root an. cond. $1.40066$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.99 − 0.171i)2-s + (−1.69 − 2.47i)3-s + (3.94 − 0.681i)4-s + (0.344 + 0.596i)5-s + (−3.79 − 4.64i)6-s + (3.20 − 5.55i)7-s + (7.73 − 2.03i)8-s + (−3.27 + 8.38i)9-s + (0.788 + 1.13i)10-s + (−2.32 + 4.03i)11-s + (−8.35 − 8.61i)12-s + (−10.7 + 6.19i)13-s + (5.43 − 11.6i)14-s + (0.895 − 1.86i)15-s + (15.0 − 5.37i)16-s + 26.4i·17-s + ⋯
L(s)  = 1  + (0.996 − 0.0855i)2-s + (−0.564 − 0.825i)3-s + (0.985 − 0.170i)4-s + (0.0689 + 0.119i)5-s + (−0.632 − 0.774i)6-s + (0.458 − 0.793i)7-s + (0.967 − 0.254i)8-s + (−0.363 + 0.931i)9-s + (0.0788 + 0.113i)10-s + (−0.211 + 0.366i)11-s + (−0.696 − 0.717i)12-s + (−0.825 + 0.476i)13-s + (0.388 − 0.829i)14-s + (0.0596 − 0.124i)15-s + (0.941 − 0.335i)16-s + 1.55i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.687 + 0.726i$
Analytic conductor: \(1.96185\)
Root analytic conductor: \(1.40066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1),\ 0.687 + 0.726i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.71279 - 0.737712i\)
\(L(\frac12)\) \(\approx\) \(1.71279 - 0.737712i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 + 0.171i)T \)
3 \( 1 + (1.69 + 2.47i)T \)
good5 \( 1 + (-0.344 - 0.596i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.20 + 5.55i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (2.32 - 4.03i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (10.7 - 6.19i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 26.4iT - 289T^{2} \)
19 \( 1 - 11.2iT - 361T^{2} \)
23 \( 1 + (1.52 - 0.882i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (11.0 - 19.2i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (27.1 + 47.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 57.9iT - 1.36e3T^{2} \)
41 \( 1 + (47.1 - 27.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-24.2 - 14.0i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (20.3 + 11.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 97.9T + 2.80e3T^{2} \)
59 \( 1 + (38.4 + 66.6i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (0.493 + 0.284i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (58.9 - 34.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 59.1iT - 5.04e3T^{2} \)
73 \( 1 - 19.6T + 5.32e3T^{2} \)
79 \( 1 + (63.2 - 109. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (40.9 - 70.9i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 46.6iT - 7.92e3T^{2} \)
97 \( 1 + (-32.6 + 56.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−14.17360072124335245457205346424, −13.02416469429080828474264186384, −12.31835247702254307084230451149, −11.14952450160073179668371618118, −10.30021769540601765009203784245, −7.87507986936943348977638845605, −6.94210462224504703308823204492, −5.68720438323804893638641194124, −4.24825996917290074199231242612, −1.93011662868662552424579189380, 2.98420530531271521027270573681, 4.88149416286357409472274723527, 5.49739965766214666558913919677, 7.12331824030165016398235284025, 8.904929655469745487947316660233, 10.34501721581933190510879779410, 11.51469385524133112146782372571, 12.14059880443616360594501778740, 13.50487251051641734948384519618, 14.75293515884909467962612330494

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