Properties

Label 2-12e2-144.85-c1-0-7
Degree $2$
Conductor $144$
Sign $0.975 - 0.218i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
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  + (−1.35 − 0.404i)2-s + (1.51 + 0.844i)3-s + (1.67 + 1.09i)4-s + (3.32 − 0.891i)5-s + (−1.70 − 1.75i)6-s + (−3.95 + 2.28i)7-s + (−1.82 − 2.16i)8-s + (1.57 + 2.55i)9-s + (−4.86 − 0.138i)10-s + (0.568 − 2.12i)11-s + (1.60 + 3.07i)12-s + (0.0174 + 0.0649i)13-s + (6.28 − 1.49i)14-s + (5.78 + 1.46i)15-s + (1.59 + 3.66i)16-s − 0.00952·17-s + ⋯
L(s)  = 1  + (−0.958 − 0.286i)2-s + (0.872 + 0.487i)3-s + (0.836 + 0.548i)4-s + (1.48 − 0.398i)5-s + (−0.696 − 0.717i)6-s + (−1.49 + 0.863i)7-s + (−0.644 − 0.764i)8-s + (0.524 + 0.851i)9-s + (−1.53 − 0.0436i)10-s + (0.171 − 0.639i)11-s + (0.462 + 0.886i)12-s + (0.00482 + 0.0180i)13-s + (1.68 − 0.399i)14-s + (1.49 + 0.377i)15-s + (0.398 + 0.917i)16-s − 0.00231·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.975 - 0.218i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.975 - 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04579 + 0.115614i\)
\(L(\frac12)\) \(\approx\) \(1.04579 + 0.115614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.404i)T \)
3 \( 1 + (-1.51 - 0.844i)T \)
good5 \( 1 + (-3.32 + 0.891i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (3.95 - 2.28i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.568 + 2.12i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.0174 - 0.0649i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 0.00952T + 17T^{2} \)
19 \( 1 + (-2.79 + 2.79i)T - 19iT^{2} \)
23 \( 1 + (4.81 + 2.78i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.28 + 0.343i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (1.30 - 2.25i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.32 + 3.32i)T + 37iT^{2} \)
41 \( 1 + (4.34 + 2.50i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.718 - 2.68i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.77 + 4.81i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.80 - 1.80i)T + 53iT^{2} \)
59 \( 1 + (-0.221 + 0.0592i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (10.8 + 2.91i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.19 - 4.45i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.21iT - 71T^{2} \)
73 \( 1 - 6.99iT - 73T^{2} \)
79 \( 1 + (-6.86 - 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.8 - 3.18i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.28iT - 89T^{2} \)
97 \( 1 + (4.69 + 8.12i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.14773904431940587579456288962, −12.22385200353848926051705096835, −10.58597270818895637513505186692, −9.673150458309692744110042880194, −9.268336431502932127130775006801, −8.454542251505777456106794696351, −6.73126878181470174957903348400, −5.65147310925568973819076976618, −3.29249275351775039859127278534, −2.19484380060433436497458759697, 1.78639548374852855251917201677, 3.24722893612396590750083301190, 6.02255750799514622508268130266, 6.75607456763155774715627442871, 7.66941173248427082381287595660, 9.247417040150085939526903666968, 9.804699799659512622907905779766, 10.33412735489900193528975755612, 12.19542286511728773686803539835, 13.37751437679765103320276524901

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