Properties

Label 2-12e2-144.85-c1-0-15
Degree $2$
Conductor $144$
Sign $0.793 - 0.607i$
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  + (1.05 + 0.937i)2-s + (1.45 − 0.944i)3-s + (0.241 + 1.98i)4-s + (−0.491 + 0.131i)5-s + (2.42 + 0.362i)6-s + (−2.40 + 1.38i)7-s + (−1.60 + 2.32i)8-s + (1.21 − 2.74i)9-s + (−0.644 − 0.321i)10-s + (1.06 − 3.96i)11-s + (2.22 + 2.65i)12-s + (−0.596 − 2.22i)13-s + (−3.84 − 0.785i)14-s + (−0.589 + 0.655i)15-s + (−3.88 + 0.959i)16-s + 2.87·17-s + ⋯
L(s)  = 1  + (0.748 + 0.663i)2-s + (0.838 − 0.545i)3-s + (0.120 + 0.992i)4-s + (−0.219 + 0.0589i)5-s + (0.989 + 0.147i)6-s + (−0.909 + 0.524i)7-s + (−0.567 + 0.823i)8-s + (0.405 − 0.913i)9-s + (−0.203 − 0.101i)10-s + (0.320 − 1.19i)11-s + (0.642 + 0.766i)12-s + (−0.165 − 0.617i)13-s + (−1.02 − 0.209i)14-s + (−0.152 + 0.169i)15-s + (−0.970 + 0.239i)16-s + 0.698·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.793 - 0.607i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.793 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69287 + 0.573660i\)
\(L(\frac12)\) \(\approx\) \(1.69287 + 0.573660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.05 - 0.937i)T \)
3 \( 1 + (-1.45 + 0.944i)T \)
good5 \( 1 + (0.491 - 0.131i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.40 - 1.38i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.06 + 3.96i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.596 + 2.22i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 2.87T + 17T^{2} \)
19 \( 1 + (3.48 - 3.48i)T - 19iT^{2} \)
23 \( 1 + (3.85 + 2.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.88 - 1.57i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (1.28 - 2.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.64 - 7.64i)T + 37iT^{2} \)
41 \( 1 + (-4.84 - 2.79i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.911 + 3.40i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (4.94 + 8.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.86 + 2.86i)T + 53iT^{2} \)
59 \( 1 + (-2.15 + 0.577i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.79 - 1.28i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-3.96 - 14.7i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 13.2iT - 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 + (1.56 + 2.71i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.0 - 2.95i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.37iT - 89T^{2} \)
97 \( 1 + (5.04 + 8.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.26149943831916301607820963026, −12.53856697318518320067666649730, −11.70130615233397677698541148116, −9.923818486095163485467219494487, −8.574603545519250307178846001302, −7.980484118100739916200279078376, −6.59476092014174639452979014074, −5.81830067944318291295362515990, −3.80112233792011621088095857283, −2.85488147958217809025541565692, 2.32857282609103003884842430248, 3.83061783629166836876130204128, 4.56758838497402539875533608152, 6.38123810455526404019879637114, 7.64801644799610698450093331014, 9.439752013125612146969748021192, 9.793727760383149169804970647707, 10.92129521702028984029228079076, 12.20868074831495118458477427125, 13.01966130279748785155491724604

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