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Properties

Label 2-12e2-144.133-c1-0-15
Degree $2$
Conductor $144$
Sign $0.900 - 0.434i$
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  + (1.41 + 0.0512i)2-s + (0.543 + 1.64i)3-s + (1.99 + 0.144i)4-s + (0.430 − 1.60i)5-s + (0.683 + 2.35i)6-s + (−3.62 − 2.09i)7-s + (2.81 + 0.307i)8-s + (−2.40 + 1.78i)9-s + (0.690 − 2.24i)10-s + (−4.63 + 1.24i)11-s + (0.845 + 3.35i)12-s + (3.28 + 0.879i)13-s + (−5.01 − 3.14i)14-s + (2.87 − 0.164i)15-s + (3.95 + 0.578i)16-s − 2.14·17-s + ⋯
L(s)  = 1  + (0.999 + 0.0362i)2-s + (0.313 + 0.949i)3-s + (0.997 + 0.0724i)4-s + (0.192 − 0.718i)5-s + (0.279 + 0.960i)6-s + (−1.37 − 0.791i)7-s + (0.994 + 0.108i)8-s + (−0.803 + 0.595i)9-s + (0.218 − 0.710i)10-s + (−1.39 + 0.374i)11-s + (0.244 + 0.969i)12-s + (0.910 + 0.243i)13-s + (−1.34 − 0.840i)14-s + (0.742 − 0.0425i)15-s + (0.989 + 0.144i)16-s − 0.519·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.900 - 0.434i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.900 - 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83594 + 0.419657i\)
\(L(\frac12)\) \(\approx\) \(1.83594 + 0.419657i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0512i)T \)
3 \( 1 + (-0.543 - 1.64i)T \)
good5 \( 1 + (-0.430 + 1.60i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (3.62 + 2.09i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.63 - 1.24i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.28 - 0.879i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 2.14T + 17T^{2} \)
19 \( 1 + (-1.03 + 1.03i)T - 19iT^{2} \)
23 \( 1 + (-0.405 + 0.234i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.75 + 6.55i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-3.18 - 5.50i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.728 + 0.728i)T + 37iT^{2} \)
41 \( 1 + (2.52 - 1.45i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.6 + 2.84i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (4.61 - 7.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.17 - 1.17i)T + 53iT^{2} \)
59 \( 1 + (-0.397 + 1.48i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.01 + 7.53i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (9.82 + 2.63i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.27iT - 71T^{2} \)
73 \( 1 - 8.16iT - 73T^{2} \)
79 \( 1 + (-3.63 + 6.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.03 - 7.59i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 11.1iT - 89T^{2} \)
97 \( 1 + (5.67 - 9.83i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.34767517182999540075504505813, −12.55132235390275377559202460647, −11.00753499378399195417504554847, −10.28528010715759998646379556716, −9.229617106253713999669916512335, −7.81294946355988498030086082387, −6.38723304918740994395834877574, −5.14517719355047747089809349437, −4.08641212073027789706809475500, −2.87699363983037632456249618391, 2.55073462482283165653896239200, 3.28101322713458873300653689210, 5.66300148162215228910183443714, 6.33221787317775351279279192503, 7.34959138088008056822914861101, 8.676604145325025238379042937564, 10.23905174931796291102461212353, 11.24721330518008121874526498158, 12.43502153448909609304916207960, 13.13370599334371201056767094081

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