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Properties

Label 1-588-588.11-r0-0-0
Degree $1$
Conductor $588$
Sign $0.325 + 0.945i$
Analytic cond. $2.73066$
Root an. cond. $2.73066$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.733 + 0.680i)5-s + (0.826 + 0.563i)11-s + (−0.900 + 0.433i)13-s + (−0.365 + 0.930i)17-s + (0.5 − 0.866i)19-s + (0.365 + 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.623 − 0.781i)29-s + (0.5 + 0.866i)31-s + (−0.988 + 0.149i)37-s + (0.222 − 0.974i)41-s + (0.222 + 0.974i)43-s + (0.0747 − 0.997i)47-s + (0.988 + 0.149i)53-s + (0.222 + 0.974i)55-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)5-s + (0.826 + 0.563i)11-s + (−0.900 + 0.433i)13-s + (−0.365 + 0.930i)17-s + (0.5 − 0.866i)19-s + (0.365 + 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.623 − 0.781i)29-s + (0.5 + 0.866i)31-s + (−0.988 + 0.149i)37-s + (0.222 − 0.974i)41-s + (0.222 + 0.974i)43-s + (0.0747 − 0.997i)47-s + (0.988 + 0.149i)53-s + (0.222 + 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.325 + 0.945i$
Analytic conductor: \(2.73066\)
Root analytic conductor: \(2.73066\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 588,\ (0:\ ),\ 0.325 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.229053761 + 0.8770058257i\)
\(L(\frac12)\) \(\approx\) \(1.229053761 + 0.8770058257i\)
\(L(1)\) \(\approx\) \(1.149866929 + 0.3154206827i\)
\(L(1)\) \(\approx\) \(1.149866929 + 0.3154206827i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.733 + 0.680i)T \)
11 \( 1 + (0.826 + 0.563i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (-0.365 + 0.930i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.365 + 0.930i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.988 + 0.149i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (0.0747 - 0.997i)T \)
53 \( 1 + (0.988 + 0.149i)T \)
59 \( 1 + (-0.733 + 0.680i)T \)
61 \( 1 + (-0.988 + 0.149i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.0747 + 0.997i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.69418812828410551532389445036, −22.29411675049841940007785726384, −21.295319675346026355505303180272, −20.490919343270335172631790553253, −19.87041941652380032392910378281, −18.77601518131532089832187535613, −17.97005292190424716015187130364, −16.929869660931384784931738050953, −16.62006112706466549302151128360, −15.47852286578612612584931212153, −14.338333255759724609675234671361, −13.8357862706404494136627177264, −12.72956969114638885403051123789, −12.097449858515133348538099088846, −11.02665662268585500911858314743, −9.91909360522819609541714031024, −9.2625840101633848816361790327, −8.40089836364885902784039692682, −7.27092936246148571852532559840, −6.21747323092719428555696146598, −5.32905303131903465040498303048, −4.46771098010168264338809635665, −3.15702237090858229377696092384, −2.01922135987787190478599665128, −0.79832829835634942645336847904, 1.53779320174573806424427516101, 2.434779687949493894122509840986, 3.60824360150101857364683860128, 4.74756361219813997592937988097, 5.80449419877286118017984281190, 6.82238863035206380421005497118, 7.362768172929087843068849091484, 8.86932310646321314280627636638, 9.57723989664702674872329398522, 10.383600114910409164110112604261, 11.38300996507776012256923042, 12.21631823380869483389240450570, 13.31723742320336355793349828226, 14.05704672838703479871440231257, 14.90144957652459724000009792421, 15.5460658560597829570145731586, 17.05784886431443054689809234838, 17.32811089820661488511175330161, 18.22140862304077435674968567233, 19.360312019399977236908985931974, 19.752417043833837069107787168071, 21.062325166877602930180684833076, 21.73823437070725749463822019208, 22.38517151389501282968742360777, 23.16208989118168410846121455539

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