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Properties

Label 2-12e2-144.59-c1-0-4
Degree $2$
Conductor $144$
Sign $0.195 - 0.980i$
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  + (1.00 + 0.991i)2-s + (−1.49 − 0.874i)3-s + (0.0333 + 1.99i)4-s + (0.473 + 1.76i)5-s + (−0.639 − 2.36i)6-s + (1.40 + 2.43i)7-s + (−1.94 + 2.04i)8-s + (1.46 + 2.61i)9-s + (−1.27 + 2.25i)10-s + (1.55 − 5.79i)11-s + (1.69 − 3.01i)12-s + (0.296 + 1.10i)13-s + (−0.997 + 3.85i)14-s + (0.837 − 3.05i)15-s + (−3.99 + 0.133i)16-s − 0.699i·17-s + ⋯
L(s)  = 1  + (0.712 + 0.701i)2-s + (−0.863 − 0.505i)3-s + (0.0166 + 0.999i)4-s + (0.211 + 0.789i)5-s + (−0.261 − 0.965i)6-s + (0.531 + 0.920i)7-s + (−0.689 + 0.724i)8-s + (0.489 + 0.871i)9-s + (−0.403 + 0.711i)10-s + (0.468 − 1.74i)11-s + (0.490 − 0.871i)12-s + (0.0823 + 0.307i)13-s + (−0.266 + 1.02i)14-s + (0.216 − 0.788i)15-s + (−0.999 + 0.0333i)16-s − 0.169i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.195 - 0.980i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.195 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.997649 + 0.818600i\)
\(L(\frac12)\) \(\approx\) \(0.997649 + 0.818600i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.00 - 0.991i)T \)
3 \( 1 + (1.49 + 0.874i)T \)
good5 \( 1 + (-0.473 - 1.76i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.40 - 2.43i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.55 + 5.79i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.296 - 1.10i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 0.699iT - 17T^{2} \)
19 \( 1 + (2.01 + 2.01i)T + 19iT^{2} \)
23 \( 1 + (4.91 + 2.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.45 + 5.41i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-5.15 - 2.97i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.48 - 3.48i)T + 37iT^{2} \)
41 \( 1 + (1.55 - 2.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.09 - 1.90i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.83 - 4.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.45 + 4.45i)T - 53iT^{2} \)
59 \( 1 + (13.6 - 3.65i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (13.2 + 3.54i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (12.8 - 3.45i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.21iT - 71T^{2} \)
73 \( 1 + 3.75iT - 73T^{2} \)
79 \( 1 + (-2.96 + 1.71i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.15 - 0.308i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 0.391T + 89T^{2} \)
97 \( 1 + (0.875 + 1.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.57923343957696002497166761510, −12.21293405745055698926541868060, −11.58283568078158110984895880940, −10.72923576969136117472688589998, −8.797466899053780906874352489698, −7.82539216935023350569343184702, −6.21871850472945098201596633378, −6.16922402459724367083330594757, −4.61400936383794108108023551561, −2.70368774537123496205284083461, 1.45183075546738423454201354321, 4.11846470542251452880681627691, 4.68322963704223288809842814759, 5.90370117372484364949203780839, 7.27659968109142976203759493322, 9.229080007317201623830903128735, 10.14525626913087559930885077863, 10.83892698144822733221067376555, 12.14490250730460731986489047419, 12.49334806784813900843855891246

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