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Properties

Label 2-12e2-48.35-c3-0-18
Degree $2$
Conductor $144$
Sign $-0.772 + 0.635i$
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  + (−1.64 − 2.30i)2-s + (−2.59 + 7.56i)4-s + (−7.28 − 7.28i)5-s + 29.9·7-s + (21.6 − 6.48i)8-s + (−4.78 + 28.7i)10-s + (−0.408 + 0.408i)11-s + (−26.6 − 26.6i)13-s + (−49.2 − 68.9i)14-s + (−50.5 − 39.2i)16-s − 83.0i·17-s + (−51.6 + 51.6i)19-s + (74.0 − 36.2i)20-s + (1.61 + 0.268i)22-s − 173. i·23-s + ⋯
L(s)  = 1  + (−0.581 − 0.813i)2-s + (−0.323 + 0.946i)4-s + (−0.651 − 0.651i)5-s + 1.61·7-s + (0.958 − 0.286i)8-s + (−0.151 + 0.909i)10-s + (−0.0111 + 0.0111i)11-s + (−0.568 − 0.568i)13-s + (−0.940 − 1.31i)14-s + (−0.790 − 0.613i)16-s − 1.18i·17-s + (−0.623 + 0.623i)19-s + (0.827 − 0.405i)20-s + (0.0156 + 0.00259i)22-s − 1.57i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.635i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.772 + 0.635i$
Analytic conductor: \(8.49627\)
Root analytic conductor: \(2.91483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :3/2),\ -0.772 + 0.635i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.326569 - 0.911379i\)
\(L(\frac12)\) \(\approx\) \(0.326569 - 0.911379i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.64 + 2.30i)T \)
3 \( 1 \)
good5 \( 1 + (7.28 + 7.28i)T + 125iT^{2} \)
7 \( 1 - 29.9T + 343T^{2} \)
11 \( 1 + (0.408 - 0.408i)T - 1.33e3iT^{2} \)
13 \( 1 + (26.6 + 26.6i)T + 2.19e3iT^{2} \)
17 \( 1 + 83.0iT - 4.91e3T^{2} \)
19 \( 1 + (51.6 - 51.6i)T - 6.85e3iT^{2} \)
23 \( 1 + 173. iT - 1.21e4T^{2} \)
29 \( 1 + (167. - 167. i)T - 2.43e4iT^{2} \)
31 \( 1 + 191. iT - 2.97e4T^{2} \)
37 \( 1 + (-185. + 185. i)T - 5.06e4iT^{2} \)
41 \( 1 + 62.7T + 6.89e4T^{2} \)
43 \( 1 + (193. + 193. i)T + 7.95e4iT^{2} \)
47 \( 1 - 93.1T + 1.03e5T^{2} \)
53 \( 1 + (-249. - 249. i)T + 1.48e5iT^{2} \)
59 \( 1 + (24.6 - 24.6i)T - 2.05e5iT^{2} \)
61 \( 1 + (-451. - 451. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-453. + 453. i)T - 3.00e5iT^{2} \)
71 \( 1 + 348. iT - 3.57e5T^{2} \)
73 \( 1 + 923. iT - 3.89e5T^{2} \)
79 \( 1 - 989. iT - 4.93e5T^{2} \)
83 \( 1 + (325. + 325. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.00e3T + 7.04e5T^{2} \)
97 \( 1 - 997.T + 9.12e5T^{2} \)
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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

−12.05612061503166371414888192708, −11.26860250140082754817032231688, −10.37037389761666759124705035824, −8.987716017229798490361900550627, −8.167915796619046690067322236999, −7.43591019537891582473529820616, −5.06649902067553322601822700210, −4.16512410753108974029640834966, −2.24779651198385421749370857700, −0.60042468287977219289227478295, 1.72898719835802668435869398756, 4.19654981539181889349636353610, 5.39490838524879223826090090246, 6.87609379292420968301726841587, 7.77435325822312417311521281455, 8.521054107730257954589177367069, 9.857370457336718950289259765183, 11.10885743518518754933549983533, 11.51242660239798537634439502780, 13.29831767633148311677888000812

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