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Properties

Label 2-72-72.11-c1-0-6
Degree $2$
Conductor $72$
Sign $0.704 + 0.709i$
Analytic cond. $0.574922$
Root an. cond. $0.758236$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.12 + 0.862i)2-s + (0.418 − 1.68i)3-s + (0.511 − 1.93i)4-s + (−1.60 − 2.78i)5-s + (0.980 + 2.24i)6-s + (1.82 + 1.05i)7-s + (1.09 + 2.60i)8-s + (−2.64 − 1.40i)9-s + (4.20 + 1.73i)10-s + (3.47 + 2.00i)11-s + (−3.03 − 1.66i)12-s + (−0.341 + 0.197i)13-s + (−2.94 + 0.392i)14-s + (−5.35 + 1.53i)15-s + (−3.47 − 1.97i)16-s + 1.20i·17-s + ⋯
L(s)  = 1  + (−0.792 + 0.609i)2-s + (0.241 − 0.970i)3-s + (0.255 − 0.966i)4-s + (−0.719 − 1.24i)5-s + (0.400 + 0.916i)6-s + (0.688 + 0.397i)7-s + (0.386 + 0.922i)8-s + (−0.883 − 0.469i)9-s + (1.33 + 0.548i)10-s + (1.04 + 0.605i)11-s + (−0.876 − 0.481i)12-s + (−0.0948 + 0.0547i)13-s + (−0.788 + 0.105i)14-s + (−1.38 + 0.397i)15-s + (−0.869 − 0.494i)16-s + 0.292i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.704 + 0.709i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ 0.704 + 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.622182 - 0.259016i\)
\(L(\frac12)\) \(\approx\) \(0.622182 - 0.259016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 - 0.862i)T \)
3 \( 1 + (-0.418 + 1.68i)T \)
good5 \( 1 + (1.60 + 2.78i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.82 - 1.05i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.47 - 2.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.341 - 0.197i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.20iT - 17T^{2} \)
19 \( 1 + 1.62T + 19T^{2} \)
23 \( 1 + (-2.74 - 4.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.95 + 5.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.34 + 1.93i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 + (1.23 - 0.715i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.21 - 2.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.792 + 1.37i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.07T + 53T^{2} \)
59 \( 1 + (2.29 - 1.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.18 + 4.72i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.60 + 4.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.69T + 71T^{2} \)
73 \( 1 - 9.49T + 73T^{2} \)
79 \( 1 + (-1.53 - 0.886i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.30 + 0.755i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.2iT - 89T^{2} \)
97 \( 1 + (-5.84 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.76443171591892906531114388408, −13.50027935289146691638728829602, −12.12769226139601447743213715056, −11.50974830539407792799566364488, −9.451555866011288059287021841224, −8.486589654666224307559450998802, −7.79855447360569161300075210335, −6.41350412703383057733337968514, −4.80328813773384434714049220641, −1.45928096222127515576478818258, 3.03718913967386118541571283882, 4.22135047638566990136498126202, 6.82991345986324474356725257639, 8.119380102612132859704638094277, 9.190396908295398497939537176916, 10.71377736334231792761570862817, 10.92554107906035083042504973035, 12.05301158337697153316303626418, 14.05476594723919265809422397691, 14.76995308394917264781304323831

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