↑

Properties

Label 2-588-28.3-c1-0-7
Degree $2$
Conductor $588$
Sign $0.834 - 0.550i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
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  + (0.630 − 1.26i)2-s + (−0.5 + 0.866i)3-s + (−1.20 − 1.59i)4-s + (−1.46 + 0.848i)5-s + (0.780 + 1.17i)6-s + (−2.78 + 0.516i)8-s + (−0.499 − 0.866i)9-s + (0.146 + 2.39i)10-s + (2.61 + 1.51i)11-s + (1.98 − 0.244i)12-s + 6.04i·13-s − 1.69i·15-s + (−1.09 + 3.84i)16-s + (3.76 + 2.17i)17-s + (−1.41 + 0.0865i)18-s + (−0.561 − 0.972i)19-s + ⋯
L(s)  = 1  + (0.446 − 0.895i)2-s + (−0.288 + 0.499i)3-s + (−0.602 − 0.798i)4-s + (−0.656 + 0.379i)5-s + (0.318 + 0.481i)6-s + (−0.983 + 0.182i)8-s + (−0.166 − 0.288i)9-s + (0.0464 + 0.757i)10-s + (0.788 + 0.455i)11-s + (0.573 − 0.0705i)12-s + 1.67i·13-s − 0.437i·15-s + (−0.274 + 0.961i)16-s + (0.912 + 0.526i)17-s + (−0.332 + 0.0204i)18-s + (−0.128 − 0.223i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.834 - 0.550i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.834 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14220 + 0.342572i\)
\(L(\frac12)\) \(\approx\) \(1.14220 + 0.342572i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.630 + 1.26i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (1.46 - 0.848i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.61 - 1.51i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.04iT - 13T^{2} \)
17 \( 1 + (-3.76 - 2.17i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.561 + 0.972i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.61 + 1.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.56 - 6.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.73iT - 41T^{2} \)
43 \( 1 - 8.10iT - 43T^{2} \)
47 \( 1 + (5.12 + 8.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.12 + 3.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.16 - 4.71i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.79 - 1.03i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + (-2.93 - 1.69i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.08 - 2.35i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.24T + 83T^{2} \)
89 \( 1 + (6.70 - 3.86i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.68iT - 97T^{2} \)
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

−11.09314023673408654856915335989, −9.915523054932174796980233809604, −9.426474225289652499498200557127, −8.390827200818459251803825875296, −6.96781737151501293542438489830, −6.13684658546503361403803792661, −4.79267177434008550546378432743, −4.10937958237731453123338832790, −3.20290782584221720984592164431, −1.58682540260592006995785928185, 0.65012582877287655327773126415, 3.07896598928470252315693370359, 4.07844195589967697979503407375, 5.35853650420503085191946129557, 5.90538614138266140526614918057, 7.13345973801352804960382996011, 7.79871399881673922547822596453, 8.488759591510143040915685211097, 9.486906061658635448457490984726, 10.76412319063662641434438981434

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