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Properties

Label 2-12e2-144.13-c1-0-10
Degree $2$
Conductor $144$
Sign $0.682 + 0.730i$
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  + (−0.595 − 1.28i)2-s + (1.66 − 0.460i)3-s + (−1.28 + 1.52i)4-s + (0.722 + 2.69i)5-s + (−1.58 − 1.86i)6-s + (2.89 − 1.67i)7-s + (2.72 + 0.743i)8-s + (2.57 − 1.53i)9-s + (3.02 − 2.53i)10-s + (−4.60 − 1.23i)11-s + (−1.44 + 3.14i)12-s + (−1.48 + 0.398i)13-s + (−3.86 − 2.71i)14-s + (2.44 + 4.17i)15-s + (−0.673 − 3.94i)16-s − 6.47·17-s + ⋯
L(s)  = 1  + (−0.421 − 0.906i)2-s + (0.963 − 0.265i)3-s + (−0.644 + 0.764i)4-s + (0.323 + 1.20i)5-s + (−0.647 − 0.762i)6-s + (1.09 − 0.631i)7-s + (0.964 + 0.262i)8-s + (0.858 − 0.512i)9-s + (0.957 − 0.801i)10-s + (−1.38 − 0.371i)11-s + (−0.418 + 0.908i)12-s + (−0.412 + 0.110i)13-s + (−1.03 − 0.725i)14-s + (0.632 + 1.07i)15-s + (−0.168 − 0.985i)16-s − 1.57·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.682 + 0.730i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.682 + 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11750 - 0.485420i\)
\(L(\frac12)\) \(\approx\) \(1.11750 - 0.485420i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.595 + 1.28i)T \)
3 \( 1 + (-1.66 + 0.460i)T \)
good5 \( 1 + (-0.722 - 2.69i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.89 + 1.67i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.60 + 1.23i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.48 - 0.398i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 6.47T + 17T^{2} \)
19 \( 1 + (-0.957 - 0.957i)T + 19iT^{2} \)
23 \( 1 + (-3.70 - 2.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.289 + 1.07i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-1.89 + 3.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.14 - 6.14i)T - 37iT^{2} \)
41 \( 1 + (5.04 + 2.91i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.31 + 1.69i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.81 - 3.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.762 + 0.762i)T - 53iT^{2} \)
59 \( 1 + (-2.17 - 8.11i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.80 - 6.74i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.81 + 0.487i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.98iT - 71T^{2} \)
73 \( 1 + 5.45iT - 73T^{2} \)
79 \( 1 + (2.95 + 5.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.77 + 10.3i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 2.24iT - 89T^{2} \)
97 \( 1 + (-6.50 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.35349666182455639964939456926, −11.72485192479468678999535122004, −10.71282547210547321247355403501, −10.19196199022818448301145383887, −8.808953182847890002177023706037, −7.83507116644387005368075916832, −7.02313773041097845695932654361, −4.68443790541944690756169899226, −3.13287495127833957897578925584, −2.07482511505663195585231449736, 2.03487701293062148527152137107, 4.84704327277187061088064594465, 5.06244128560243838328848926642, 7.13806084011237834003571825331, 8.403317798097159725133354000854, 8.642609026827351885836886946087, 9.736128940866767776256321121849, 10.87162365311823058619009702703, 12.70641603364890045462508167611, 13.36426574578258450028042741536

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