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Properties

Label 2-588-21.2-c2-0-3
Degree $2$
Conductor $588$
Sign $-0.991 - 0.126i$
Analytic cond. $16.0218$
Root an. cond. $4.00272$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 2.59i)3-s + (−4.5 + 7.79i)9-s − 22·13-s + (−13 + 22.5i)19-s + (−12.5 − 21.6i)25-s − 27·27-s + (23 + 39.8i)31-s + (−13 + 22.5i)37-s + (−33 − 57.1i)39-s − 22·43-s − 78·57-s + (−37 + 64.0i)61-s + (−61 − 105. i)67-s + (23 + 39.8i)73-s + (37.5 − 64.9i)75-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)9-s − 1.69·13-s + (−0.684 + 1.18i)19-s + (−0.5 − 0.866i)25-s − 27-s + (0.741 + 1.28i)31-s + (−0.351 + 0.608i)37-s + (−0.846 − 1.46i)39-s − 0.511·43-s − 1.36·57-s + (−0.606 + 1.05i)61-s + (−0.910 − 1.57i)67-s + (0.315 + 0.545i)73-s + (0.5 − 0.866i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ -0.991 - 0.126i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9230457527\)
\(L(\frac12)\) \(\approx\) \(0.9230457527\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 - 2.59i)T \)
7 \( 1 \)
good5 \( 1 + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 + 22T + 169T^{2} \)
17 \( 1 + (144.5 - 250. i)T^{2} \)
19 \( 1 + (13 - 22.5i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (264.5 + 458. i)T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + (-23 - 39.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (13 - 22.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 22T + 1.84e3T^{2} \)
47 \( 1 + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (37 - 64.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (61 + 105. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + (-23 - 39.8i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-71 + 122. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 2T + 9.40e3T^{2} \)
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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

−10.46585338978960106948698527716, −10.17092957099179829229765038957, −9.251823007726348740135108796656, −8.318516202391227996821782381265, −7.58279288877201456951900338993, −6.34266505967319717829527087006, −5.12273017027325649984311164230, −4.38849390343872713736224695249, −3.20254896310358304500820519513, −2.09510523340834369967397111383, 0.29796106070280674850463531125, 2.01958346474810853171097571869, 2.91445358431442396938335986775, 4.34929001111113580725694286479, 5.52673322623685321328878916635, 6.70286024441232755281635274247, 7.36565784553269006973898591406, 8.177906159312076729411681894531, 9.204219592854962955375168092537, 9.830187807622395318128480603650

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