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Properties

Label 2-72-72.11-c1-0-0
Degree $2$
Conductor $72$
Sign $0.667 - 0.744i$
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  + (−1.40 + 0.111i)2-s + (−1.71 − 0.231i)3-s + (1.97 − 0.313i)4-s + (1.74 + 3.01i)5-s + (2.44 + 0.136i)6-s + (1.80 + 1.04i)7-s + (−2.75 + 0.660i)8-s + (2.89 + 0.795i)9-s + (−2.79 − 4.06i)10-s + (−0.116 − 0.0675i)11-s + (−3.46 + 0.0795i)12-s + (−2.63 + 1.52i)13-s + (−2.66 − 1.27i)14-s + (−2.29 − 5.58i)15-s + (3.80 − 1.23i)16-s − 4.19i·17-s + ⋯
L(s)  = 1  + (−0.996 + 0.0785i)2-s + (−0.990 − 0.133i)3-s + (0.987 − 0.156i)4-s + (0.779 + 1.35i)5-s + (0.998 + 0.0556i)6-s + (0.683 + 0.394i)7-s + (−0.972 + 0.233i)8-s + (0.964 + 0.265i)9-s + (−0.883 − 1.28i)10-s + (−0.0352 − 0.0203i)11-s + (−0.999 + 0.0229i)12-s + (−0.731 + 0.422i)13-s + (−0.712 − 0.339i)14-s + (−0.591 − 1.44i)15-s + (0.950 − 0.309i)16-s − 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.667 - 0.744i$
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: \((2,\ 72,\ (\ :1/2),\ 0.667 - 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.502630 + 0.224450i\)
\(L(\frac12)\) \(\approx\) \(0.502630 + 0.224450i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.111i)T \)
3 \( 1 + (1.71 + 0.231i)T \)
good5 \( 1 + (-1.74 - 3.01i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.80 - 1.04i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.116 + 0.0675i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.63 - 1.52i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.19iT - 17T^{2} \)
19 \( 1 - 0.919T + 19T^{2} \)
23 \( 1 + (-0.689 - 1.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.24 + 7.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.39 - 2.53i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.61iT - 37T^{2} \)
41 \( 1 + (-1.79 + 1.03i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.41 + 9.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.205 - 0.356i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.968T + 53T^{2} \)
59 \( 1 + (-3.88 + 2.24i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.44 + 4.29i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.15 - 5.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + 4.06T + 73T^{2} \)
79 \( 1 + (-10.8 - 6.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.23 - 3.02i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.35iT - 89T^{2} \)
97 \( 1 + (0.477 - 0.826i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.97088758574809157250808700697, −13.92835048467479007784212461946, −12.06751468291407635884503497985, −11.26572364968157138743612111986, −10.36984997378611447026363969090, −9.426376328608089773669044063070, −7.52370383338204410020095903893, −6.65495640800549514971201894843, −5.44839105589224587843496515525, −2.28951442477191674827549885237, 1.35701781222327843385092819165, 4.80829609516533304167432200730, 6.03107002593195687650788738258, 7.63354824851523704054865937090, 8.944653791497725485395748542111, 10.03390696439833679762292967525, 10.94373155242871265975861597831, 12.24385270547313969971813024573, 12.93588000341063768635184315643, 14.75886721933899907486941920718

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