↑

Properties

Label 2-54e2-108.79-c0-0-5
Degree $2$
Conductor $2916$
Sign $0.893 - 0.448i$
Analytic cond. $1.45527$
Root an. cond. $1.20634$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (1.43 − 1.20i)5-s + (0.5 − 0.866i)8-s + (0.939 + 1.62i)10-s + (0.0603 + 0.342i)13-s + (0.766 + 0.642i)16-s + (0.766 + 1.32i)17-s + (−1.76 + 0.642i)20-s + (0.439 − 2.49i)25-s − 0.347·26-s + (−0.0603 + 0.342i)29-s + (−0.766 + 0.642i)32-s + (−1.43 + 0.524i)34-s + (−0.766 − 1.32i)37-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (1.43 − 1.20i)5-s + (0.5 − 0.866i)8-s + (0.939 + 1.62i)10-s + (0.0603 + 0.342i)13-s + (0.766 + 0.642i)16-s + (0.766 + 1.32i)17-s + (−1.76 + 0.642i)20-s + (0.439 − 2.49i)25-s − 0.347·26-s + (−0.0603 + 0.342i)29-s + (−0.766 + 0.642i)32-s + (−1.43 + 0.524i)34-s + (−0.766 − 1.32i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $0.893 - 0.448i$
Analytic conductor: \(1.45527\)
Root analytic conductor: \(1.20634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2916} (2107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2916,\ (\ :0),\ 0.893 - 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.379230258\)
\(L(\frac12)\) \(\approx\) \(1.379230258\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 \)
good5 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.766 + 0.642i)T^{2} \)
11 \( 1 + (-0.173 - 0.984i)T^{2} \)
13 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.766 - 0.642i)T^{2} \)
37 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.939 + 0.342i)T^{2} \)
89 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.871315603425364431668000145330, −8.421748472164334592357469412488, −7.54424492121732707245886397617, −6.53694086326522849695485843234, −5.88758979637459525628739067887, −5.39717176980925236814165766845, −4.62517653694581457481169502890, −3.71406554222988343173851369926, −2.02217789616964619430663697965, −1.14466595618812650956480862040, 1.30073823267768278368139415200, 2.44762847287396107678297769936, 2.88898221318386701775022858483, 3.83249429324775027341993994821, 5.17847686267821035462871269516, 5.59836463124048457987885267840, 6.67159341062982669575543572978, 7.35512573518808113652259792993, 8.288095127358512526208637854878, 9.360253753913521435076993114606

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