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Properties

Label 2-12e2-144.13-c1-0-20
Degree $2$
Conductor $144$
Sign $-0.458 + 0.888i$
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  + (0.562 − 1.29i)2-s + (0.388 − 1.68i)3-s + (−1.36 − 1.46i)4-s + (0.226 + 0.846i)5-s + (−1.97 − 1.45i)6-s + (0.567 − 0.327i)7-s + (−2.66 + 0.950i)8-s + (−2.69 − 1.31i)9-s + (1.22 + 0.182i)10-s + (5.75 + 1.54i)11-s + (−2.99 + 1.73i)12-s + (−4.44 + 1.19i)13-s + (−0.105 − 0.919i)14-s + (1.51 − 0.0535i)15-s + (−0.266 + 3.99i)16-s + 2.75·17-s + ⋯
L(s)  = 1  + (0.398 − 0.917i)2-s + (0.224 − 0.974i)3-s + (−0.683 − 0.730i)4-s + (0.101 + 0.378i)5-s + (−0.804 − 0.593i)6-s + (0.214 − 0.123i)7-s + (−0.941 + 0.335i)8-s + (−0.899 − 0.437i)9-s + (0.387 + 0.0576i)10-s + (1.73 + 0.464i)11-s + (−0.865 + 0.501i)12-s + (−1.23 + 0.330i)13-s + (−0.0282 − 0.245i)14-s + (0.391 − 0.0138i)15-s + (−0.0667 + 0.997i)16-s + 0.668·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.458 + 0.888i$
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: \((2,\ 144,\ (\ :1/2),\ -0.458 + 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.702390 - 1.15252i\)
\(L(\frac12)\) \(\approx\) \(0.702390 - 1.15252i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.562 + 1.29i)T \)
3 \( 1 + (-0.388 + 1.68i)T \)
good5 \( 1 + (-0.226 - 0.846i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.567 + 0.327i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.75 - 1.54i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (4.44 - 1.19i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 2.75T + 17T^{2} \)
19 \( 1 + (1.73 + 1.73i)T + 19iT^{2} \)
23 \( 1 + (-3.50 - 2.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.662 - 2.47i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-2.08 + 3.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.30 - 4.30i)T - 37iT^{2} \)
41 \( 1 + (6.15 + 3.55i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.841 + 0.225i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.65 - 8.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.64 - 7.64i)T - 53iT^{2} \)
59 \( 1 + (1.83 + 6.83i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.01 + 3.77i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (11.6 - 3.11i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 4.34iT - 71T^{2} \)
73 \( 1 - 0.656iT - 73T^{2} \)
79 \( 1 + (8.16 + 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.43 - 5.36i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 5.11iT - 89T^{2} \)
97 \( 1 + (-3.05 - 5.29i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.49568258872476364408039436139, −12.02123857915310104915720460378, −11.05938159944463222080496904360, −9.699237226266568022363611138185, −8.855095642834876005264949452938, −7.26312015556846920900759043534, −6.28943065911088116420572078875, −4.65756352119124002295812655037, −3.05927705244324971721608182437, −1.56980043856854100641270915314, 3.36013875664827886000245817875, 4.58891227366880815064008462007, 5.55126249748472620214976236147, 6.91167545644478605407678435783, 8.375822054344302613544464486564, 9.072767443976735125562295216932, 10.06411283813730226089641168638, 11.62319719534354671710157197763, 12.47949672817346031633161985814, 13.82615774576967491417937594741

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