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Properties

Label 2-72-72.11-c1-0-8
Degree $2$
Conductor $72$
Sign $-0.0128 + 0.999i$
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  + (0.186 − 1.40i)2-s + (0.418 − 1.68i)3-s + (−1.93 − 0.523i)4-s + (1.60 + 2.78i)5-s + (−2.27 − 0.900i)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 + (−1.68 + 3.02i)12-s + (0.341 − 0.197i)13-s + (−1.81 + 2.35i)14-s + (5.35 − 1.53i)15-s + (3.45 + 2.02i)16-s + 1.20i·17-s + ⋯
L(s)  = 1  + (0.132 − 0.991i)2-s + (0.241 − 0.970i)3-s + (−0.965 − 0.261i)4-s + (0.719 + 1.24i)5-s + (−0.929 − 0.367i)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 + (−0.487 + 0.873i)12-s + (0.0948 − 0.0547i)13-s + (−0.485 + 0.630i)14-s + (1.38 − 0.397i)15-s + (0.862 + 0.505i)16-s + 0.292i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.0128 + 0.999i$
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: \((2,\ 72,\ (\ :1/2),\ -0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.698707 - 0.707759i\)
\(L(\frac12)\) \(\approx\) \(0.698707 - 0.707759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.186 + 1.40i)T \)
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

−14.31869720605212893287099243355, −13.15719534030385594029492997268, −12.34660958628558377417255351176, −11.05408796833931441802802494143, −10.06189489100026288681445055867, −8.921973091538967345697587913023, −7.08396045837329015440925329341, −6.09604931963359637617391915362, −3.58297790741058513013486155578, −2.10734416985607204939599599251, 3.84087732465043202255617100790, 5.25126877548436741070106933373, 6.20662500580077710268226757241, 8.318775129180915008105381154200, 9.225151376455100801530525660272, 9.751988621253314932949373698111, 11.81682419115256999962390995474, 13.19275989080261570340854398232, 13.89776642967543030649985482564, 15.11555088275243338813039169568

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