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Properties

Label 2-12e3-72.13-c1-0-20
Degree $2$
Conductor $1728$
Sign $-0.495 + 0.868i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
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  + (0.476 − 0.275i)11-s − 7.89·17-s − 6.34i·19-s + (−2.5 − 4.33i)25-s + (−3.39 + 5.88i)41-s + (10.6 − 6.17i)43-s + (3.5 − 6.06i)49-s + (−13.2 − 7.62i)59-s + (−0.301 − 0.174i)67-s − 15.6·73-s + (15.5 − 9i)83-s − 18·89-s + (−4.84 − 8.39i)97-s − 14.1i·107-s + (9 − 15.5i)113-s + ⋯
L(s)  = 1  + (0.143 − 0.0829i)11-s − 1.91·17-s − 1.45i·19-s + (−0.5 − 0.866i)25-s + (−0.530 + 0.919i)41-s + (1.63 − 0.941i)43-s + (0.5 − 0.866i)49-s + (−1.71 − 0.992i)59-s + (−0.0368 − 0.0212i)67-s − 1.83·73-s + (1.71 − 0.987i)83-s − 1.90·89-s + (−0.492 − 0.852i)97-s − 1.36i·107-s + (0.846 − 1.46i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.495 + 0.868i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.495 + 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8813984840\)
\(L(\frac12)\) \(\approx\) \(0.8813984840\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.476 + 0.275i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.89T + 17T^{2} \)
19 \( 1 + 6.34iT - 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (3.39 - 5.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.6 + 6.17i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (13.2 + 7.62i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.301 + 0.174i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-15.5 + 9i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + (4.84 + 8.39i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.971947247240765674545621962750, −8.462156726116416767094342212804, −7.35621597029727009762091338748, −6.69243612399491475888182183832, −5.93191498404799664475062795979, −4.76272231378331561772780390561, −4.20869948854724981757043647138, −2.90592525436959118530643204182, −1.99813797290488288558334319903, −0.31891392844759892390505726419, 1.51937619692144916135724562997, 2.59201883081624651051863911295, 3.82904029194184710066628038965, 4.50893208763552433007829134451, 5.64735549088642500233628087454, 6.32657668750413825304488605258, 7.25490499626604958309279626756, 7.964417022279569383144926019752, 8.934833922634731036423470464638, 9.402521567561272438353763328965

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