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Properties

Label 2-12e2-144.131-c1-0-1
Degree $2$
Conductor $144$
Sign $-0.751 - 0.659i$
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  + (−1.19 + 0.763i)2-s + (0.0841 + 1.73i)3-s + (0.835 − 1.81i)4-s + (−1.17 + 0.315i)5-s + (−1.42 − 1.99i)6-s + (1.93 + 3.35i)7-s + (0.392 + 2.80i)8-s + (−2.98 + 0.291i)9-s + (1.16 − 1.27i)10-s + (−2.53 − 0.678i)11-s + (3.21 + 1.29i)12-s + (−2.21 + 0.594i)13-s + (−4.86 − 2.51i)14-s + (−0.645 − 2.01i)15-s + (−2.60 − 3.03i)16-s + 1.65i·17-s + ⋯
L(s)  = 1  + (−0.841 + 0.539i)2-s + (0.0485 + 0.998i)3-s + (0.417 − 0.908i)4-s + (−0.527 + 0.141i)5-s + (−0.579 − 0.814i)6-s + (0.732 + 1.26i)7-s + (0.138 + 0.990i)8-s + (−0.995 + 0.0970i)9-s + (0.367 − 0.403i)10-s + (−0.763 − 0.204i)11-s + (0.927 + 0.372i)12-s + (−0.615 + 0.164i)13-s + (−1.30 − 0.672i)14-s + (−0.166 − 0.519i)15-s + (−0.651 − 0.758i)16-s + 0.401i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.751 - 0.659i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.232907 + 0.618160i\)
\(L(\frac12)\) \(\approx\) \(0.232907 + 0.618160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 - 0.763i)T \)
3 \( 1 + (-0.0841 - 1.73i)T \)
good5 \( 1 + (1.17 - 0.315i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.93 - 3.35i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.53 + 0.678i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.21 - 0.594i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 1.65iT - 17T^{2} \)
19 \( 1 + (-2.32 + 2.32i)T - 19iT^{2} \)
23 \( 1 + (-6.27 - 3.62i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.23 - 1.40i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-6.44 - 3.72i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.499 + 0.499i)T - 37iT^{2} \)
41 \( 1 + (-5.22 + 9.04i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.07 + 7.72i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.91 - 3.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.69 + 4.69i)T + 53iT^{2} \)
59 \( 1 + (-1.60 - 5.98i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.01 - 7.52i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-3.58 - 13.3i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + 10.4iT - 73T^{2} \)
79 \( 1 + (7.83 - 4.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.746 + 2.78i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 4.91T + 89T^{2} \)
97 \( 1 + (7.00 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.93221377610119733690488210163, −12.01577871889084394816223314746, −11.24731983069546904619669584932, −10.33449972452750724930516895765, −9.150902651550163840003164435667, −8.495980267605048967402182566052, −7.40642482083502220475877269308, −5.67279341748657205285672687929, −4.89515169475939868567945508983, −2.69993957789130575308045979522, 0.897307741402226691609561730906, 2.76398002972602057901822390928, 4.57455740342891378970587018759, 6.73538727505621696908910066692, 7.86204156958618620652547744843, 8.006005176351647698913217709622, 9.708367452808993232928358035792, 10.81993823592528447829388604541, 11.61705167595184679814887315611, 12.54916771838016762109603792906

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