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Properties

Label 2-12e2-144.11-c1-0-0
Degree $2$
Conductor $144$
Sign $-0.984 - 0.175i$
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  + (−1.20 + 0.739i)2-s + (−0.679 − 1.59i)3-s + (0.905 − 1.78i)4-s + (−2.39 − 0.642i)5-s + (1.99 + 1.41i)6-s + (−1.93 + 3.34i)7-s + (0.228 + 2.81i)8-s + (−2.07 + 2.16i)9-s + (3.36 − 0.999i)10-s + (−4.01 + 1.07i)11-s + (−3.45 − 0.230i)12-s + (3.17 + 0.850i)13-s + (−0.147 − 5.46i)14-s + (0.605 + 4.25i)15-s + (−2.36 − 3.22i)16-s − 1.33i·17-s + ⋯
L(s)  = 1  + (−0.852 + 0.523i)2-s + (−0.392 − 0.919i)3-s + (0.452 − 0.891i)4-s + (−1.07 − 0.287i)5-s + (0.815 + 0.578i)6-s + (−0.730 + 1.26i)7-s + (0.0809 + 0.996i)8-s + (−0.692 + 0.721i)9-s + (1.06 − 0.316i)10-s + (−1.21 + 0.324i)11-s + (−0.997 − 0.0664i)12-s + (0.880 + 0.235i)13-s + (−0.0394 − 1.45i)14-s + (0.156 + 1.09i)15-s + (−0.590 − 0.807i)16-s − 0.322i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.984 - 0.175i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.984 - 0.175i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00248259 + 0.0280425i\)
\(L(\frac12)\) \(\approx\) \(0.00248259 + 0.0280425i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 - 0.739i)T \)
3 \( 1 + (0.679 + 1.59i)T \)
good5 \( 1 + (2.39 + 0.642i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.93 - 3.34i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.01 - 1.07i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.17 - 0.850i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 1.33iT - 17T^{2} \)
19 \( 1 + (6.09 + 6.09i)T + 19iT^{2} \)
23 \( 1 + (-0.521 + 0.301i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.272 + 0.0730i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (5.84 - 3.37i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.00346 - 0.00346i)T + 37iT^{2} \)
41 \( 1 + (0.614 + 1.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.151 + 0.563i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.24 + 2.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.24 - 3.24i)T - 53iT^{2} \)
59 \( 1 + (-1.44 + 5.40i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.528 - 1.97i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.65 - 9.90i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.85iT - 71T^{2} \)
73 \( 1 + 7.41iT - 73T^{2} \)
79 \( 1 + (0.839 + 0.484i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.171 + 0.639i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + (3.24 - 5.62i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.31924249696361242015575397099, −12.49884966291711636875038218552, −11.51092991079593558258970420971, −10.67145502072332622776783157347, −8.989096046165504235989633303161, −8.365165584458141731061125242121, −7.27586318323244873740895758501, −6.27470424118374001699011867805, −5.12510641048681582101693789344, −2.46228724836530983931570670800, 0.03615152167066825982388299659, 3.42076763024483427348667252175, 4.04491256893190018566470045919, 6.22114813492872675044535766371, 7.60784416307467983304007063610, 8.455050244280649567063697949623, 9.896683365805842014421964963974, 10.72922466242397888003682658114, 11.01452038423507176635087602780, 12.38839809280038021095882637386

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