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Properties

Label 2-12e2-16.5-c3-0-15
Degree $2$
Conductor $144$
Sign $0.682 - 0.730i$
Analytic cond. $8.49627$
Root an. cond. $2.91483$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.81 + 0.253i)2-s + (7.87 + 1.42i)4-s + (4.66 + 4.66i)5-s + 24.8i·7-s + (21.8 + 6.00i)8-s + (11.9 + 14.3i)10-s + (−22.3 − 22.3i)11-s + (−11.2 + 11.2i)13-s + (−6.30 + 70.1i)14-s + (59.9 + 22.4i)16-s + 88.4·17-s + (37.8 − 37.8i)19-s + (30.0 + 43.3i)20-s + (−57.2 − 68.5i)22-s + 48.1i·23-s + ⋯
L(s)  = 1  + (0.995 + 0.0894i)2-s + (0.983 + 0.178i)4-s + (0.417 + 0.417i)5-s + 1.34i·7-s + (0.964 + 0.265i)8-s + (0.378 + 0.452i)10-s + (−0.612 − 0.612i)11-s + (−0.240 + 0.240i)13-s + (−0.120 + 1.33i)14-s + (0.936 + 0.350i)16-s + 1.26·17-s + (0.456 − 0.456i)19-s + (0.336 + 0.484i)20-s + (−0.555 − 0.664i)22-s + 0.436i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.682 - 0.730i$
Analytic conductor: \(8.49627\)
Root analytic conductor: \(2.91483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :3/2),\ 0.682 - 0.730i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.95992 + 1.28591i\)
\(L(\frac12)\) \(\approx\) \(2.95992 + 1.28591i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.81 - 0.253i)T \)
3 \( 1 \)
good5 \( 1 + (-4.66 - 4.66i)T + 125iT^{2} \)
7 \( 1 - 24.8iT - 343T^{2} \)
11 \( 1 + (22.3 + 22.3i)T + 1.33e3iT^{2} \)
13 \( 1 + (11.2 - 11.2i)T - 2.19e3iT^{2} \)
17 \( 1 - 88.4T + 4.91e3T^{2} \)
19 \( 1 + (-37.8 + 37.8i)T - 6.85e3iT^{2} \)
23 \( 1 - 48.1iT - 1.21e4T^{2} \)
29 \( 1 + (10.4 - 10.4i)T - 2.43e4iT^{2} \)
31 \( 1 + 96.9T + 2.97e4T^{2} \)
37 \( 1 + (163. + 163. i)T + 5.06e4iT^{2} \)
41 \( 1 + 360. iT - 6.89e4T^{2} \)
43 \( 1 + (100. + 100. i)T + 7.95e4iT^{2} \)
47 \( 1 + 220.T + 1.03e5T^{2} \)
53 \( 1 + (-175. - 175. i)T + 1.48e5iT^{2} \)
59 \( 1 + (405. + 405. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-664. + 664. i)T - 2.26e5iT^{2} \)
67 \( 1 + (107. - 107. i)T - 3.00e5iT^{2} \)
71 \( 1 - 215. iT - 3.57e5T^{2} \)
73 \( 1 - 668. iT - 3.89e5T^{2} \)
79 \( 1 + 822.T + 4.93e5T^{2} \)
83 \( 1 + (326. - 326. i)T - 5.71e5iT^{2} \)
89 \( 1 + 262. iT - 7.04e5T^{2} \)
97 \( 1 + 150.T + 9.12e5T^{2} \)
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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

−12.71326160789917663391728117410, −11.97833207139023083184465210217, −11.00029459203721119600673995787, −9.835641527104766958235102321617, −8.442990356240700621827574602814, −7.16794950317229579683253579876, −5.84048980071961413444050216315, −5.25943177819263370540955664910, −3.33576527910975409698491109702, −2.22123199727115955838857709083, 1.38194445328200651859342101976, 3.26837500823397515924816028818, 4.60792403722599336583053059627, 5.61218539933164399154308641889, 7.06301536099586387350973454997, 7.87567724355601315011605055184, 9.881706644709431972212270007346, 10.41970854742383168867791012586, 11.69409212959015853697890197418, 12.76819739618919602744910474062

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