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Properties

Label 2-12e2-144.85-c1-0-18
Degree $2$
Conductor $144$
Sign $0.600 + 0.799i$
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  + (1.10 − 0.877i)2-s + (1.71 − 0.241i)3-s + (0.459 − 1.94i)4-s + (−1.98 + 0.531i)5-s + (1.69 − 1.77i)6-s + (−1.54 + 0.894i)7-s + (−1.19 − 2.56i)8-s + (2.88 − 0.828i)9-s + (−1.73 + 2.33i)10-s + (−0.693 + 2.58i)11-s + (0.318 − 3.44i)12-s + (1.24 + 4.63i)13-s + (−0.933 + 2.35i)14-s + (−3.27 + 1.39i)15-s + (−3.57 − 1.79i)16-s − 3.58·17-s + ⋯
L(s)  = 1  + (0.784 − 0.620i)2-s + (0.990 − 0.139i)3-s + (0.229 − 0.973i)4-s + (−0.887 + 0.237i)5-s + (0.690 − 0.723i)6-s + (−0.585 + 0.338i)7-s + (−0.423 − 0.905i)8-s + (0.961 − 0.276i)9-s + (−0.548 + 0.737i)10-s + (−0.208 + 0.779i)11-s + (0.0920 − 0.995i)12-s + (0.344 + 1.28i)13-s + (−0.249 + 0.628i)14-s + (−0.845 + 0.359i)15-s + (−0.894 − 0.447i)16-s − 0.870·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.600 + 0.799i$
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: \((2,\ 144,\ (\ :1/2),\ 0.600 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62478 - 0.811155i\)
\(L(\frac12)\) \(\approx\) \(1.62478 - 0.811155i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.10 + 0.877i)T \)
3 \( 1 + (-1.71 + 0.241i)T \)
good5 \( 1 + (1.98 - 0.531i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.54 - 0.894i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.693 - 2.58i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.24 - 4.63i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 3.58T + 17T^{2} \)
19 \( 1 + (-4.85 + 4.85i)T - 19iT^{2} \)
23 \( 1 + (-0.446 - 0.257i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.44 + 1.72i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-4.05 + 7.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.25 - 1.25i)T + 37iT^{2} \)
41 \( 1 + (4.07 + 2.35i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.76 - 6.57i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (3.48 + 6.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.26 - 5.26i)T + 53iT^{2} \)
59 \( 1 + (-6.76 + 1.81i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.78 - 1.55i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.453 + 1.69i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.58iT - 71T^{2} \)
73 \( 1 + 12.5iT - 73T^{2} \)
79 \( 1 + (4.01 + 6.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.99 - 2.14i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 + (4.15 + 7.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.18426080737758587353458655069, −11.94826532173803393538037512983, −11.29206012404361657511304153162, −9.758813210727565135715436483653, −9.100124083978507504880363992885, −7.47197346937692687521524370316, −6.55451479690390765768117461262, −4.59735676068047863917156719445, −3.58438105600173420159698716905, −2.26349353580981858582762600140, 3.16903770138178009038367906547, 3.87336101440067275872850648406, 5.41874571274969931864524004274, 6.97145724807087197893870279770, 7.998833996579913769121121804790, 8.596226683063939656468905051729, 10.14252777753313186473783358509, 11.46228743676069350240938758892, 12.69344559467494021939960235959, 13.32035310658099963716480559875

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