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Properties

Label 2-72-72.61-c1-0-2
Degree $2$
Conductor $72$
Sign $0.340 - 0.940i$
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  + (1.05 + 0.947i)2-s + (−1.52 + 0.816i)3-s + (0.206 + 1.98i)4-s + (0.602 − 0.348i)5-s + (−2.37 − 0.589i)6-s + (0.795 − 1.37i)7-s + (−1.66 + 2.28i)8-s + (1.66 − 2.49i)9-s + (0.962 + 0.205i)10-s + (−2.37 − 1.36i)11-s + (−1.93 − 2.87i)12-s + (4.76 − 2.75i)13-s + (2.13 − 0.693i)14-s + (−0.636 + 1.02i)15-s + (−3.91 + 0.819i)16-s − 5.65·17-s + ⋯
L(s)  = 1  + (0.742 + 0.669i)2-s + (−0.882 + 0.471i)3-s + (0.103 + 0.994i)4-s + (0.269 − 0.155i)5-s + (−0.970 − 0.240i)6-s + (0.300 − 0.520i)7-s + (−0.589 + 0.807i)8-s + (0.555 − 0.831i)9-s + (0.304 + 0.0649i)10-s + (−0.715 − 0.412i)11-s + (−0.559 − 0.828i)12-s + (1.32 − 0.763i)13-s + (0.571 − 0.185i)14-s + (−0.164 + 0.264i)15-s + (−0.978 + 0.204i)16-s − 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.340 - 0.940i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ 0.340 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.876756 + 0.615058i\)
\(L(\frac12)\) \(\approx\) \(0.876756 + 0.615058i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.05 - 0.947i)T \)
3 \( 1 + (1.52 - 0.816i)T \)
good5 \( 1 + (-0.602 + 0.348i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.795 + 1.37i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.37 + 1.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.76 + 2.75i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
19 \( 1 + 0.963iT - 19T^{2} \)
23 \( 1 + (-3.28 - 5.69i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.85 + 1.64i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.69 + 6.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.25iT - 37T^{2} \)
41 \( 1 + (0.931 + 1.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.99 - 1.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.85 + 6.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.54iT - 53T^{2} \)
59 \( 1 + (-4.62 + 2.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.93 + 4.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.95 - 3.43i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 - 2.83T + 73T^{2} \)
79 \( 1 + (-2.87 + 4.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.74 - 3.31i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.98T + 89T^{2} \)
97 \( 1 + (1.24 - 2.16i)T + (-48.5 - 84.0i)T^{2} \)
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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

−15.28242739982870885838407159512, −13.45646279190138685522672517242, −13.14578747561769722429590889640, −11.44488170842381860726591431650, −10.81905963104551486153312750507, −9.066119929165669989863583963070, −7.57985470382888312724949018932, −6.14571756175975980283027333988, −5.19106950389860951369404351704, −3.76707217298851479536211762504, 2.08195674293928121016472794343, 4.49583117848105205069057465377, 5.79150577543534229142285836454, 6.83076516523778266959048123576, 8.879945469003759129364490469796, 10.57552300061442713217955365716, 11.14872030615339241758176193449, 12.32070363753068172070476034254, 13.13681548071164454976779059216, 14.11021185901510557613071515560

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