↑

Properties

Label 2-2740-2740.2359-c0-0-0
Degree $2$
Conductor $2740$
Sign $0.992 + 0.123i$
Analytic cond. $1.36743$
Root an. cond. $1.16937$
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  + (0.850 − 0.526i)2-s + (0.445 − 0.895i)4-s + (−0.932 + 0.361i)5-s + (−0.0922 − 0.995i)8-s + (0.673 + 0.739i)9-s + (−0.602 + 0.798i)10-s + (1.07 + 1.56i)13-s + (−0.602 − 0.798i)16-s + (−0.183 + 1.98i)17-s + (0.961 + 0.273i)18-s + (−0.0922 + 0.995i)20-s + (0.739 − 0.673i)25-s + (1.73 + 0.765i)26-s + (0.227 − 1.63i)29-s + (−0.932 − 0.361i)32-s + ⋯
L(s)  = 1  + (0.850 − 0.526i)2-s + (0.445 − 0.895i)4-s + (−0.932 + 0.361i)5-s + (−0.0922 − 0.995i)8-s + (0.673 + 0.739i)9-s + (−0.602 + 0.798i)10-s + (1.07 + 1.56i)13-s + (−0.602 − 0.798i)16-s + (−0.183 + 1.98i)17-s + (0.961 + 0.273i)18-s + (−0.0922 + 0.995i)20-s + (0.739 − 0.673i)25-s + (1.73 + 0.765i)26-s + (0.227 − 1.63i)29-s + (−0.932 − 0.361i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2740\)    =    \(2^{2} \cdot 5 \cdot 137\)
Sign: $0.992 + 0.123i$
Analytic conductor: \(1.36743\)
Root analytic conductor: \(1.16937\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2740} (2359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2740,\ (\ :0),\ 0.992 + 0.123i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.842170489\)
\(L(\frac12)\) \(\approx\) \(1.842170489\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.850 + 0.526i)T \)
5 \( 1 + (0.932 - 0.361i)T \)
137 \( 1 + (0.982 - 0.183i)T \)
good3 \( 1 + (-0.673 - 0.739i)T^{2} \)
7 \( 1 + (-0.932 - 0.361i)T^{2} \)
11 \( 1 + (-0.602 - 0.798i)T^{2} \)
13 \( 1 + (-1.07 - 1.56i)T + (-0.361 + 0.932i)T^{2} \)
17 \( 1 + (0.183 - 1.98i)T + (-0.982 - 0.183i)T^{2} \)
19 \( 1 + (-0.850 - 0.526i)T^{2} \)
23 \( 1 + (0.961 + 0.273i)T^{2} \)
29 \( 1 + (-0.227 + 1.63i)T + (-0.961 - 0.273i)T^{2} \)
31 \( 1 + (-0.526 - 0.850i)T^{2} \)
37 \( 1 - 1.05T + T^{2} \)
41 \( 1 + (0.688 + 0.688i)T + iT^{2} \)
43 \( 1 + (-0.526 + 0.850i)T^{2} \)
47 \( 1 + (0.995 + 0.0922i)T^{2} \)
53 \( 1 + (0.621 + 1.11i)T + (-0.526 + 0.850i)T^{2} \)
59 \( 1 + (-0.0922 + 0.995i)T^{2} \)
61 \( 1 + (0.811 - 0.890i)T + (-0.0922 - 0.995i)T^{2} \)
67 \( 1 + (-0.361 + 0.932i)T^{2} \)
71 \( 1 + (-0.798 - 0.602i)T^{2} \)
73 \( 1 + (0.247 + 1.32i)T + (-0.932 + 0.361i)T^{2} \)
79 \( 1 + (-0.673 + 0.739i)T^{2} \)
83 \( 1 + (0.183 + 0.982i)T^{2} \)
89 \( 1 + (-1.78 + 0.418i)T + (0.895 - 0.445i)T^{2} \)
97 \( 1 + (1.34 - 0.449i)T + (0.798 - 0.602i)T^{2} \)
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

−8.993207656188875507469366705049, −8.169049365918891713688350052286, −7.38544561224889578972429506209, −6.45571618256381200353003019235, −6.07348047434994350926291078088, −4.72011922110967390853598187099, −4.08106930502610650940427224224, −3.71370878317001639973376469551, −2.33227223657552974650998370868, −1.49513025592771257179769312462, 1.02617065575553195945067492284, 2.96406549781276061761031939400, 3.40606103909141051831090280351, 4.39182457613957977937650793531, 5.04613177581585167110326427083, 5.86514225922758842264687177429, 6.85221546736998139512146435593, 7.35171285951788054256826626893, 8.110613798928852084997526080920, 8.776810292330093439754850492709

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