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Properties

Label 2-12e2-144.83-c1-0-7
Degree $2$
Conductor $144$
Sign $0.301 + 0.953i$
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  + (−1.25 − 0.649i)2-s + (−1.73 − 0.0841i)3-s + (1.15 + 1.63i)4-s + (−0.315 + 1.17i)5-s + (2.11 + 1.22i)6-s + (1.93 − 3.35i)7-s + (−0.392 − 2.80i)8-s + (2.98 + 0.291i)9-s + (1.16 − 1.27i)10-s + (−0.678 − 2.53i)11-s + (−1.86 − 2.92i)12-s + (0.594 − 2.21i)13-s + (−4.61 + 2.95i)14-s + (0.645 − 2.01i)15-s + (−1.32 + 3.77i)16-s − 1.65i·17-s + ⋯
L(s)  = 1  + (−0.888 − 0.459i)2-s + (−0.998 − 0.0485i)3-s + (0.578 + 0.815i)4-s + (−0.141 + 0.527i)5-s + (0.864 + 0.501i)6-s + (0.732 − 1.26i)7-s + (−0.138 − 0.990i)8-s + (0.995 + 0.0970i)9-s + (0.367 − 0.403i)10-s + (−0.204 − 0.763i)11-s + (−0.537 − 0.843i)12-s + (0.164 − 0.615i)13-s + (−1.23 + 0.790i)14-s + (0.166 − 0.519i)15-s + (−0.331 + 0.943i)16-s − 0.401i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.301 + 0.953i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.301 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.460284 - 0.337024i\)
\(L(\frac12)\) \(\approx\) \(0.460284 - 0.337024i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 + 0.649i)T \)
3 \( 1 + (1.73 + 0.0841i)T \)
good5 \( 1 + (0.315 - 1.17i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.93 + 3.35i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.678 + 2.53i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.594 + 2.21i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + 1.65iT - 17T^{2} \)
19 \( 1 + (-2.32 + 2.32i)T - 19iT^{2} \)
23 \( 1 + (-6.27 + 3.62i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.40 - 5.23i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (6.44 - 3.72i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.499 + 0.499i)T - 37iT^{2} \)
41 \( 1 + (5.22 + 9.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.72 - 2.07i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.91 - 3.31i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.69 - 4.69i)T + 53iT^{2} \)
59 \( 1 + (-5.98 - 1.60i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-7.52 + 2.01i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (13.3 + 3.58i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + 10.4iT - 73T^{2} \)
79 \( 1 + (-7.83 - 4.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.78 + 0.746i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 4.91T + 89T^{2} \)
97 \( 1 + (7.00 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.65836850238393200579737651648, −11.38116346942354201204104637411, −10.84043191914321018360119943217, −10.34044641357996519703906010759, −8.771608419164531763176210058979, −7.40373418187158049180458803629, −6.86607077171684006747958247220, −5.04861607820757037041038242020, −3.37979338923431600644154375148, −0.957373425283540113686391845240, 1.71953261301195850922188038220, 4.85669329543764481984088939530, 5.64092197642193329749511685107, 6.89717588636641194776572215828, 8.118525021631414064690902342793, 9.151859087076970207313550487131, 10.11193020436390534399929096789, 11.45046176658147434697325896453, 11.83881420843249565222010770939, 13.05893339524991720896575503927

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