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Properties

Label 2-12e2-16.5-c1-0-2
Degree $2$
Conductor $144$
Sign $0.995 + 0.0985i$
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  + (−0.635 − 1.26i)2-s + (−1.19 + 1.60i)4-s + (2.68 + 2.68i)5-s + 2.15i·7-s + (2.78 + 0.484i)8-s + (1.68 − 5.09i)10-s + (−1.79 − 1.79i)11-s + (1.38 − 1.38i)13-s + (2.72 − 1.37i)14-s + (−1.15 − 3.82i)16-s + 0.224·17-s + (0.158 − 0.158i)19-s + (−7.51 + 1.11i)20-s + (−1.12 + 3.41i)22-s − 2.82i·23-s + ⋯
L(s)  = 1  + (−0.449 − 0.893i)2-s + (−0.595 + 0.803i)4-s + (1.20 + 1.20i)5-s + 0.816i·7-s + (0.985 + 0.171i)8-s + (0.533 − 1.61i)10-s + (−0.542 − 0.542i)11-s + (0.383 − 0.383i)13-s + (0.728 − 0.366i)14-s + (−0.289 − 0.957i)16-s + 0.0545·17-s + (0.0364 − 0.0364i)19-s + (−1.68 + 0.248i)20-s + (−0.240 + 0.727i)22-s − 0.589i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.995 + 0.0985i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.995 + 0.0985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.988651 - 0.0488537i\)
\(L(\frac12)\) \(\approx\) \(0.988651 - 0.0488537i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.635 + 1.26i)T \)
3 \( 1 \)
good5 \( 1 + (-2.68 - 2.68i)T + 5iT^{2} \)
7 \( 1 - 2.15iT - 7T^{2} \)
11 \( 1 + (1.79 + 1.79i)T + 11iT^{2} \)
13 \( 1 + (-1.38 + 1.38i)T - 13iT^{2} \)
17 \( 1 - 0.224T + 17T^{2} \)
19 \( 1 + (-0.158 + 0.158i)T - 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (-1.85 + 1.85i)T - 29iT^{2} \)
31 \( 1 - 1.84T + 31T^{2} \)
37 \( 1 + (3.66 + 3.66i)T + 37iT^{2} \)
41 \( 1 - 5.88iT - 41T^{2} \)
43 \( 1 + (7.75 + 7.75i)T + 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (7.51 + 7.51i)T + 53iT^{2} \)
59 \( 1 + (4 + 4i)T + 59iT^{2} \)
61 \( 1 + (-5.98 + 5.98i)T - 61iT^{2} \)
67 \( 1 + (10.4 - 10.4i)T - 67iT^{2} \)
71 \( 1 - 4.31iT - 71T^{2} \)
73 \( 1 + 5.97iT - 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + (-10.1 + 10.1i)T - 83iT^{2} \)
89 \( 1 - 1.42iT - 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−13.10575916266574914379031434457, −11.90246057491278787755478866249, −10.81082814891378839453619657686, −10.22813261097075529553223745077, −9.196909474388241560149565684940, −8.106127225537083581865927370094, −6.56867505302786400303207575549, −5.36796836401240533052893647351, −3.21493914555029767279109406586, −2.21564323290093609887249339154, 1.44294800074729286752610383688, 4.51524986262941988309783634583, 5.46365753711928377767426301603, 6.63673421171652998052724894093, 7.897124704953180929746473887672, 8.985092694909070913866599196708, 9.788416087770481297434997852198, 10.63124530722775633865338608493, 12.43158725545409155951399294832, 13.59403653979803023238579678498

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