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Properties

Label 2-12e2-144.83-c1-0-10
Degree $2$
Conductor $144$
Sign $0.853 + 0.520i$
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  + (−0.639 + 1.26i)2-s + (−1.65 + 0.501i)3-s + (−1.18 − 1.61i)4-s + (0.997 − 3.72i)5-s + (0.427 − 2.41i)6-s + (−0.481 + 0.833i)7-s + (2.79 − 0.458i)8-s + (2.49 − 1.66i)9-s + (4.05 + 3.63i)10-s + (−1.00 − 3.75i)11-s + (2.76 + 2.08i)12-s + (−0.430 + 1.60i)13-s + (−0.743 − 1.14i)14-s + (0.213 + 6.67i)15-s + (−1.20 + 3.81i)16-s − 2.58i·17-s + ⋯
L(s)  = 1  + (−0.452 + 0.891i)2-s + (−0.957 + 0.289i)3-s + (−0.590 − 0.806i)4-s + (0.445 − 1.66i)5-s + (0.174 − 0.984i)6-s + (−0.181 + 0.315i)7-s + (0.986 − 0.162i)8-s + (0.832 − 0.554i)9-s + (1.28 + 1.15i)10-s + (−0.303 − 1.13i)11-s + (0.799 + 0.601i)12-s + (−0.119 + 0.445i)13-s + (−0.198 − 0.304i)14-s + (0.0550 + 1.72i)15-s + (−0.301 + 0.953i)16-s − 0.626i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.853 + 0.520i$
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: \((2,\ 144,\ (\ :1/2),\ 0.853 + 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.586412 - 0.164779i\)
\(L(\frac12)\) \(\approx\) \(0.586412 - 0.164779i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.639 - 1.26i)T \)
3 \( 1 + (1.65 - 0.501i)T \)
good5 \( 1 + (-0.997 + 3.72i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.481 - 0.833i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.00 + 3.75i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.430 - 1.60i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + 2.58iT - 17T^{2} \)
19 \( 1 + (-4.02 + 4.02i)T - 19iT^{2} \)
23 \( 1 + (-0.600 + 0.346i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.54 + 5.75i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (3.07 - 1.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.11 - 2.11i)T - 37iT^{2} \)
41 \( 1 + (-4.97 - 8.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.65 + 1.78i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.88 - 5.00i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.68 - 6.68i)T + 53iT^{2} \)
59 \( 1 + (8.75 + 2.34i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.12 - 0.570i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-4.20 - 1.12i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 9.40iT - 71T^{2} \)
73 \( 1 + 3.77iT - 73T^{2} \)
79 \( 1 + (-9.52 - 5.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.79 + 2.35i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + (-1.76 + 3.05i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.15252709611617450456089006304, −12.02390795245876119628814719381, −10.89834428776125784738061525826, −9.402682089910609123462670816261, −9.168641753729954584986454678876, −7.75716212650148670062974035367, −6.21140449961220368690469356171, −5.39581390673432235651064767851, −4.58044151253643202928736705952, −0.839844519932463917497269834795, 2.06611215214752081948453013776, 3.68627954323308610635379785362, 5.50397685491900314187001823461, 7.00021794473088958081237855668, 7.59886436476070911718128025896, 9.667609352865335420261710522109, 10.42874572559828675016906524916, 10.86879028839198381849704480537, 12.05480709987016113233516893313, 12.87107873860256211500701562255

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