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Properties

Label 2-12e2-16.13-c1-0-4
Degree $2$
Conductor $144$
Sign $0.773 - 0.633i$
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  + (1.34 + 0.443i)2-s + (1.60 + 1.19i)4-s + (−1.27 + 1.27i)5-s + 0.158i·7-s + (1.62 + 2.31i)8-s + (−2.27 + 1.14i)10-s + (3.79 − 3.79i)11-s + (−4.21 − 4.21i)13-s + (−0.0705 + 0.213i)14-s + (1.15 + 3.82i)16-s − 3.05·17-s + (−2.15 − 2.15i)19-s + (−3.55 + 0.526i)20-s + (6.78 − 3.41i)22-s + 2.82i·23-s + ⋯
L(s)  = 1  + (0.949 + 0.313i)2-s + (0.803 + 0.595i)4-s + (−0.568 + 0.568i)5-s + 0.0600i·7-s + (0.575 + 0.817i)8-s + (−0.718 + 0.361i)10-s + (1.14 − 1.14i)11-s + (−1.16 − 1.16i)13-s + (−0.0188 + 0.0570i)14-s + (0.289 + 0.957i)16-s − 0.740·17-s + (−0.495 − 0.495i)19-s + (−0.795 + 0.117i)20-s + (1.44 − 0.727i)22-s + 0.589i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.773 - 0.633i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.773 - 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61475 + 0.577244i\)
\(L(\frac12)\) \(\approx\) \(1.61475 + 0.577244i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 - 0.443i)T \)
3 \( 1 \)
good5 \( 1 + (1.27 - 1.27i)T - 5iT^{2} \)
7 \( 1 - 0.158iT - 7T^{2} \)
11 \( 1 + (-3.79 + 3.79i)T - 11iT^{2} \)
13 \( 1 + (4.21 + 4.21i)T + 13iT^{2} \)
17 \( 1 + 3.05T + 17T^{2} \)
19 \( 1 + (2.15 + 2.15i)T + 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (2.09 + 2.09i)T + 29iT^{2} \)
31 \( 1 - 4.15T + 31T^{2} \)
37 \( 1 + (5.98 - 5.98i)T - 37iT^{2} \)
41 \( 1 + 2.60iT - 41T^{2} \)
43 \( 1 + (-5.75 + 5.75i)T - 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (3.55 - 3.55i)T - 53iT^{2} \)
59 \( 1 + (4 - 4i)T - 59iT^{2} \)
61 \( 1 + (-3.66 - 3.66i)T + 61iT^{2} \)
67 \( 1 + (-0.767 - 0.767i)T + 67iT^{2} \)
71 \( 1 - 0.317iT - 71T^{2} \)
73 \( 1 - 1.33iT - 73T^{2} \)
79 \( 1 + 9.69T + 79T^{2} \)
83 \( 1 + (0.115 + 0.115i)T + 83iT^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + 0.571T + 97T^{2} \)
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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

−13.35583193529083687310490886959, −12.19167933819502566689317636240, −11.44082002454840278184085641700, −10.53878069512776324668785992253, −8.832556977780740870368954543238, −7.62183689279022382947963565976, −6.68169759848924165227617171727, −5.49227390299227299477929295700, −4.02538472525663143726951785769, −2.85538845104502399448770315126, 2.04231455419895829426360444600, 4.14513801824832481448778993687, 4.68214791067128009684602325879, 6.46029789153683316666679001109, 7.33633704324410486709397934067, 9.001198816080772825885274298830, 10.03395915105638759775904934388, 11.36258122250695787771610312727, 12.23081601223146809349523871637, 12.63555163242934429020980341154

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