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Properties

Label 2-588-28.27-c1-0-34
Degree $2$
Conductor $588$
Sign $-0.832 + 0.554i$
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  + (−0.856 − 1.12i)2-s + 3-s + (−0.533 + 1.92i)4-s − 3.85i·5-s + (−0.856 − 1.12i)6-s + (2.62 − 1.05i)8-s + 9-s + (−4.33 + 3.30i)10-s − 1.36i·11-s + (−0.533 + 1.92i)12-s + 0.369i·13-s − 3.85i·15-s + (−3.43 − 2.05i)16-s − 4.50i·17-s + (−0.856 − 1.12i)18-s − 0.0661·19-s + ⋯
L(s)  = 1  + (−0.605 − 0.795i)2-s + 0.577·3-s + (−0.266 + 0.963i)4-s − 1.72i·5-s + (−0.349 − 0.459i)6-s + (0.928 − 0.371i)8-s + 0.333·9-s + (−1.37 + 1.04i)10-s − 0.410i·11-s + (−0.153 + 0.556i)12-s + 0.102i·13-s − 0.995i·15-s + (−0.857 − 0.513i)16-s − 1.09i·17-s + (−0.201 − 0.265i)18-s − 0.0151·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332147 - 1.09847i\)
\(L(\frac12)\) \(\approx\) \(0.332147 - 1.09847i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.856 + 1.12i)T \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + 3.85iT - 5T^{2} \)
11 \( 1 + 1.36iT - 11T^{2} \)
13 \( 1 - 0.369iT - 13T^{2} \)
17 \( 1 + 4.50iT - 17T^{2} \)
19 \( 1 + 0.0661T + 19T^{2} \)
23 \( 1 - 3.20iT - 23T^{2} \)
29 \( 1 + 3.11T + 29T^{2} \)
31 \( 1 - 6.03T + 31T^{2} \)
37 \( 1 + 5.49T + 37T^{2} \)
41 \( 1 + 8.45iT - 41T^{2} \)
43 \( 1 + 6.30iT - 43T^{2} \)
47 \( 1 - 1.42T + 47T^{2} \)
53 \( 1 + 2.54T + 53T^{2} \)
59 \( 1 + 3.43T + 59T^{2} \)
61 \( 1 - 1.43iT - 61T^{2} \)
67 \( 1 + 9.76iT - 67T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 - 1.80iT - 73T^{2} \)
79 \( 1 - 12.4iT - 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 1.29iT - 89T^{2} \)
97 \( 1 - 2.88iT - 97T^{2} \)
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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

−10.08672276077006800213357782422, −9.296055133884961220208514253689, −8.801338936658691115373049790783, −8.085828263875865072545516468637, −7.16776122616739364278499026840, −5.41152124532899742893861025387, −4.49044958717648577296240613735, −3.45461396839785190327207004740, −2.01903806498172212038453795152, −0.75265222924768984434033176839, 1.97048130834686713745662045158, 3.22026176575951212432392716558, 4.52636982754497847590296640885, 6.08913555342620518802759262918, 6.63816386851673979255322178153, 7.54801932777287333435753458081, 8.185581610485879448421424549169, 9.286804390146993462923780304373, 10.23564042950446153577288212568, 10.57055304953561165474514943728

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