↑

Properties

Label 2-585-13.9-c1-0-23
Degree $2$
Conductor $585$
Sign $-0.662 - 0.749i$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.30 − 2.25i)2-s + (−2.38 − 4.12i)4-s − 5-s + (−1.80 − 3.11i)7-s − 7.20·8-s + (−1.30 + 2.25i)10-s + (−2.60 + 4.50i)11-s + (3.01 + 1.97i)13-s − 9.37·14-s + (−4.60 + 7.97i)16-s + (−1.46 − 2.54i)17-s + (−3.38 − 5.86i)19-s + (2.38 + 4.12i)20-s + (6.76 + 11.7i)22-s + (2.76 − 4.79i)23-s + ⋯
L(s)  = 1  + (0.919 − 1.59i)2-s + (−1.19 − 2.06i)4-s − 0.447·5-s + (−0.680 − 1.17i)7-s − 2.54·8-s + (−0.411 + 0.712i)10-s + (−0.784 + 1.35i)11-s + (0.837 + 0.547i)13-s − 2.50·14-s + (−1.15 + 1.99i)16-s + (−0.356 − 0.616i)17-s + (−0.776 − 1.34i)19-s + (0.533 + 0.923i)20-s + (1.44 + 2.49i)22-s + (0.577 − 0.999i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.662 - 0.749i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.554447 + 1.22985i\)
\(L(\frac12)\) \(\approx\) \(0.554447 + 1.22985i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
13 \( 1 + (-3.01 - 1.97i)T \)
good2 \( 1 + (-1.30 + 2.25i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (1.80 + 3.11i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.60 - 4.50i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.46 + 2.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.38 + 5.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.76 + 4.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.916 + 1.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.10T + 31T^{2} \)
37 \( 1 + (-1.76 + 3.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.68 - 4.65i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.58 + 2.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.80T + 47T^{2} \)
53 \( 1 - 5.20T + 53T^{2} \)
59 \( 1 + (3.68 + 6.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.71 - 2.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.75 - 3.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.85 + 8.40i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.805T + 73T^{2} \)
79 \( 1 + 4.10T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + (-4.91 + 8.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.78 - 4.82i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.43537252867180251154933297272, −9.748509613337287216613076837605, −8.767452738946891412862973763136, −7.23690890227924756994567334709, −6.42362568194526257187568791903, −4.68031567596640013103838403058, −4.47815315637726637485657618527, −3.28515188240909214914391705825, −2.23454026947767862374146978917, −0.55548320617604482746079748682, 3.05953480119591889173353175162, 3.75946421614812211971786566235, 5.16661243947525911268807148070, 5.96506719854977149669227443587, 6.32829312566949545356694875164, 7.70513940368002865988716992029, 8.439964158957402092111372183689, 8.829856657203603904689400245153, 10.37707238271820198015223404722, 11.53955593034454211476727105390

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