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Properties

Label 2-588-28.19-c1-0-34
Degree $2$
Conductor $588$
Sign $-0.178 + 0.983i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.850 − 1.12i)2-s + (0.5 + 0.866i)3-s + (−0.553 − 1.92i)4-s + (−0.834 − 0.481i)5-s + (1.40 + 0.171i)6-s + (−2.64 − 1.00i)8-s + (−0.499 + 0.866i)9-s + (−1.25 + 0.533i)10-s + (4.74 − 2.74i)11-s + (1.38 − 1.44i)12-s − 3.75i·13-s − 0.963i·15-s + (−3.38 + 2.12i)16-s + (0.594 − 0.343i)17-s + (0.553 + 1.30i)18-s + (2.44 − 4.22i)19-s + ⋯
L(s)  = 1  + (0.601 − 0.798i)2-s + (0.288 + 0.499i)3-s + (−0.276 − 0.960i)4-s + (−0.373 − 0.215i)5-s + (0.573 + 0.0700i)6-s + (−0.934 − 0.356i)8-s + (−0.166 + 0.288i)9-s + (−0.396 + 0.168i)10-s + (1.43 − 0.826i)11-s + (0.400 − 0.415i)12-s − 1.04i·13-s − 0.248i·15-s + (−0.846 + 0.531i)16-s + (0.144 − 0.0832i)17-s + (0.130 + 0.306i)18-s + (0.560 − 0.969i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.178 + 0.983i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.178 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26540 - 1.51636i\)
\(L(\frac12)\) \(\approx\) \(1.26540 - 1.51636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.850 + 1.12i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (0.834 + 0.481i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.74 + 2.74i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.75iT - 13T^{2} \)
17 \( 1 + (-0.594 + 0.343i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.44 + 4.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.07 + 0.620i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.48T + 29T^{2} \)
31 \( 1 + (-2.41 - 4.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.36 + 2.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.42iT - 41T^{2} \)
43 \( 1 - 5.97iT - 43T^{2} \)
47 \( 1 + (-1.80 + 3.13i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.04 - 3.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.34 - 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.01 + 5.20i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.17 + 4.71i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (-5.76 + 3.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.22 - 0.707i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.543T + 83T^{2} \)
89 \( 1 + (0.480 + 0.277i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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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

−10.58573171108376502428797681627, −9.652113998587568458225010493690, −8.959807174969593648783934229671, −8.043630260933301171518405861991, −6.58189971092631069756163587055, −5.60950410952942056112977899510, −4.56564610677369643999732228175, −3.67330017519682256375820212774, −2.79668170097378118130266529970, −0.969971460481288115072307286594, 1.91494997578510786755205305159, 3.60667340922050892722620429694, 4.19632198461237752719942913985, 5.59975566158986620339195502421, 6.61675657128999594023075558817, 7.20585617951871530179856254291, 8.009503344294558058417601954149, 9.063296580461388645858830704117, 9.705402035925717126877720573406, 11.38851977281889958138376913492

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