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Properties

Label 2-72-9.4-c1-0-2
Degree $2$
Conductor $72$
Sign $0.594 + 0.804i$
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  + (−1.18 − 1.26i)3-s + (1.68 − 2.92i)5-s + (0.686 + 1.18i)7-s + (−0.186 + 2.99i)9-s + (−0.5 − 0.866i)11-s + (−2.68 + 4.65i)13-s + (−5.68 + 1.33i)15-s + 0.372·17-s + 6.37·19-s + (0.686 − 2.27i)21-s + (−2.68 + 4.65i)23-s + (−3.18 − 5.51i)25-s + (4.00 − 3.31i)27-s + (−0.686 − 1.18i)29-s + (−0.313 + 0.543i)31-s + ⋯
L(s)  = 1  + (−0.684 − 0.728i)3-s + (0.754 − 1.30i)5-s + (0.259 + 0.449i)7-s + (−0.0620 + 0.998i)9-s + (−0.150 − 0.261i)11-s + (−0.745 + 1.29i)13-s + (−1.46 + 0.344i)15-s + 0.0902·17-s + 1.46·19-s + (0.149 − 0.496i)21-s + (−0.560 + 0.970i)23-s + (−0.637 − 1.10i)25-s + (0.769 − 0.638i)27-s + (−0.127 − 0.220i)29-s + (−0.0563 + 0.0976i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.594 + 0.804i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ 0.594 + 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753038 - 0.380035i\)
\(L(\frac12)\) \(\approx\) \(0.753038 - 0.380035i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.18 + 1.26i)T \)
good5 \( 1 + (-1.68 + 2.92i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.686 - 1.18i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.68 - 4.65i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.372T + 17T^{2} \)
19 \( 1 - 6.37T + 19T^{2} \)
23 \( 1 + (2.68 - 4.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.686 + 1.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.313 - 0.543i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.74T + 37T^{2} \)
41 \( 1 + (-0.127 + 0.221i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.87 + 8.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.686 - 1.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.68 - 2.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.87 - 6.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 5.11T + 73T^{2} \)
79 \( 1 + (-0.313 - 0.543i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.68 + 13.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-4.87 - 8.43i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.06787720123389733932250145833, −13.37231652910697514477832590145, −12.18457224441605173587741627434, −11.62816783890811019437104911187, −9.849785781643367104460999903642, −8.787589947031332386701770994099, −7.37605072270638361810659625308, −5.78935191560722170999123948828, −4.93524655292304400119499297670, −1.73332716400736499919266292061, 3.11732750306004008507032683652, 5.06172920739721406035955948384, 6.30787816442194039886959575366, 7.59561004699652374725667452262, 9.712103960320424840175102590290, 10.31658830517586373712031971696, 11.18344339303794487281191239848, 12.51524180222399523899992965535, 14.03228191793283928470283564841, 14.78383440275452899159940040580

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