Properties

Label 2-72-72.43-c2-0-10
Degree $2$
Conductor $72$
Sign $0.847 + 0.530i$
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  + (−1.95 − 0.434i)2-s + (2.76 − 1.16i)3-s + (3.62 + 1.69i)4-s + (−0.0166 + 0.00958i)5-s + (−5.90 + 1.06i)6-s + (4.07 + 2.35i)7-s + (−6.33 − 4.88i)8-s + (6.29 − 6.42i)9-s + (0.0365 − 0.0114i)10-s + (2.84 − 4.93i)11-s + (11.9 + 0.484i)12-s + (10.0 − 5.80i)13-s + (−6.92 − 6.35i)14-s + (−0.0347 + 0.0458i)15-s + (10.2 + 12.2i)16-s − 0.376·17-s + ⋯
L(s)  = 1  + (−0.976 − 0.217i)2-s + (0.921 − 0.387i)3-s + (0.905 + 0.424i)4-s + (−0.00332 + 0.00191i)5-s + (−0.984 + 0.177i)6-s + (0.581 + 0.335i)7-s + (−0.791 − 0.611i)8-s + (0.699 − 0.714i)9-s + (0.00365 − 0.00114i)10-s + (0.259 − 0.448i)11-s + (0.999 + 0.0403i)12-s + (0.773 − 0.446i)13-s + (−0.494 − 0.454i)14-s + (−0.00231 + 0.00305i)15-s + (0.639 + 0.768i)16-s − 0.0221·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.847 + 0.530i$
Analytic conductor: \(1.96185\)
Root analytic conductor: \(1.40066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1),\ 0.847 + 0.530i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.09788 - 0.315553i\)
\(L(\frac12)\) \(\approx\) \(1.09788 - 0.315553i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.95 + 0.434i)T \)
3 \( 1 + (-2.76 + 1.16i)T \)
good5 \( 1 + (0.0166 - 0.00958i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-4.07 - 2.35i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-2.84 + 4.93i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-10.0 + 5.80i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 0.376T + 289T^{2} \)
19 \( 1 + 15.0T + 361T^{2} \)
23 \( 1 + (39.1 - 22.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-32.0 - 18.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (26.3 - 15.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 53.4iT - 1.36e3T^{2} \)
41 \( 1 + (-29.0 - 50.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (23.0 - 39.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (34.2 + 19.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 0.989iT - 2.80e3T^{2} \)
59 \( 1 + (-29.4 - 50.9i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (75.1 + 43.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (34.1 + 59.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 42.3iT - 5.04e3T^{2} \)
73 \( 1 - 26.6T + 5.32e3T^{2} \)
79 \( 1 + (-121. - 69.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-40.9 + 71.0i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 42.6T + 7.92e3T^{2} \)
97 \( 1 + (-55.9 + 96.8i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−14.39963490457460990098381189070, −13.13435219235444412520404964899, −11.97947208527576190306148244227, −10.84409705846273876207647128734, −9.495641251267190489579770055152, −8.488123623618569040252697402625, −7.72760451906861385350026299452, −6.21159332628882903547810919637, −3.50073644487253294716193895779, −1.72995864400369126625561454512, 2.05263993740131768555381044687, 4.25120748390474251993752469884, 6.43675857374318151850050052862, 7.901386323261163933289446209142, 8.622083029574784878180071287612, 9.875518549466625962641996312881, 10.72301995170419359343556135198, 12.08267002114820686007634412601, 13.81592812165912472017416907470, 14.61471324185339616005434322977

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