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Properties

Label 2-12e2-144.59-c1-0-12
Degree $2$
Conductor $144$
Sign $-0.0254 + 0.999i$
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  + (−1.26 + 0.630i)2-s + (−0.377 − 1.69i)3-s + (1.20 − 1.59i)4-s + (0.282 + 1.05i)5-s + (1.54 + 1.90i)6-s + (−1.93 − 3.35i)7-s + (−0.519 + 2.78i)8-s + (−2.71 + 1.27i)9-s + (−1.02 − 1.15i)10-s + (0.946 − 3.53i)11-s + (−3.15 − 1.43i)12-s + (−1.03 − 3.87i)13-s + (4.56 + 3.02i)14-s + (1.67 − 0.874i)15-s + (−1.09 − 3.84i)16-s − 1.55i·17-s + ⋯
L(s)  = 1  + (−0.895 + 0.445i)2-s + (−0.217 − 0.975i)3-s + (0.602 − 0.798i)4-s + (0.126 + 0.471i)5-s + (0.629 + 0.776i)6-s + (−0.731 − 1.26i)7-s + (−0.183 + 0.982i)8-s + (−0.905 + 0.425i)9-s + (−0.323 − 0.365i)10-s + (0.285 − 1.06i)11-s + (−0.910 − 0.414i)12-s + (−0.288 − 1.07i)13-s + (1.21 + 0.808i)14-s + (0.432 − 0.225i)15-s + (−0.273 − 0.961i)16-s − 0.375i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0254 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0254 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.0254 + 0.999i$
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: \((2,\ 144,\ (\ :1/2),\ -0.0254 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.409722 - 0.420289i\)
\(L(\frac12)\) \(\approx\) \(0.409722 - 0.420289i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 - 0.630i)T \)
3 \( 1 + (0.377 + 1.69i)T \)
good5 \( 1 + (-0.282 - 1.05i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.93 + 3.35i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.946 + 3.53i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.03 + 3.87i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 1.55iT - 17T^{2} \)
19 \( 1 + (-4.06 - 4.06i)T + 19iT^{2} \)
23 \( 1 + (3.86 + 2.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.14 - 4.28i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (1.85 + 1.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.04 - 6.04i)T + 37iT^{2} \)
41 \( 1 + (1.59 - 2.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.51 - 1.47i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.0494 + 0.0856i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.72 + 1.72i)T - 53iT^{2} \)
59 \( 1 + (-13.3 + 3.58i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-8.69 - 2.33i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.939 - 0.251i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.11iT - 71T^{2} \)
73 \( 1 + 10.4iT - 73T^{2} \)
79 \( 1 + (14.2 - 8.22i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (15.0 + 4.02i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + (-0.0532 - 0.0922i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.94949764613391757128933211759, −11.62084903044429761868334108628, −10.64185186008170901781425830537, −9.886028399167554025773073320914, −8.347172505438874432102313153357, −7.47236033040136411838505833445, −6.60854623783511508334143162829, −5.67543255593746140760290599929, −3.05591974976526486165948921742, −0.809937019045952415702644114306, 2.44411085802718397969170696489, 4.10292921267758260897032688920, 5.65110804927275502886026786439, 7.04082415329591975824754702305, 8.745567675869483166860578228674, 9.387330936671659765572624807403, 9.893736878574956704678843241750, 11.39159308176922304484352194296, 12.01562955533871175009016041568, 12.92418247945009194936951885574

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