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Properties

Label 2-588-21.2-c2-0-1
Degree $2$
Conductor $588$
Sign $-0.696 - 0.717i$
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  + (0.447 − 2.96i)3-s + (3.24 + 1.87i)5-s + (−8.59 − 2.65i)9-s + (−15.1 + 8.77i)11-s − 4.69·13-s + (7 − 8.77i)15-s + (−19.4 + 11.2i)17-s + (−11.7 + 20.3i)19-s + (−5.5 − 9.52i)25-s + (−11.7 + 24.3i)27-s − 17.5i·29-s + (−23.4 − 40.6i)31-s + (19.2 + 49.0i)33-s + (15 − 25.9i)37-s + (−2.09 + 13.9i)39-s + ⋯
L(s)  = 1  + (0.149 − 0.988i)3-s + (0.648 + 0.374i)5-s + (−0.955 − 0.295i)9-s + (−1.38 + 0.797i)11-s − 0.360·13-s + (0.466 − 0.584i)15-s + (−1.14 + 0.660i)17-s + (−0.617 + 1.06i)19-s + (−0.220 − 0.381i)25-s + (−0.434 + 0.900i)27-s − 0.605i·29-s + (−0.756 − 1.31i)31-s + (0.582 + 1.48i)33-s + (0.405 − 0.702i)37-s + (−0.0538 + 0.356i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.696 - 0.717i$
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: \((2,\ 588,\ (\ :1),\ -0.696 - 0.717i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2213587654\)
\(L(\frac12)\) \(\approx\) \(0.2213587654\)
\(L(2)\) 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 \( 1 + (-0.447 + 2.96i)T \)
7 \( 1 \)
good5 \( 1 + (-3.24 - 1.87i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (15.1 - 8.77i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 4.69T + 169T^{2} \)
17 \( 1 + (19.4 - 11.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (11.7 - 20.3i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (264.5 + 458. i)T^{2} \)
29 \( 1 + 17.5iT - 841T^{2} \)
31 \( 1 + (23.4 + 40.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-15 + 25.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 50T + 1.84e3T^{2} \)
47 \( 1 + (-51.8 - 29.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-75.9 + 43.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (81.0 - 46.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (58.6 - 101. i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 87.7iT - 5.04e3T^{2} \)
73 \( 1 + (23.4 + 40.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-22 + 38.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 101. iT - 6.88e3T^{2} \)
89 \( 1 + (-64.8 - 37.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 75.0T + 9.40e3T^{2} \)
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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

−10.68344122206960995729990711999, −10.07851270461120070405084824830, −8.973795478699828986799503055279, −7.965923288954104907022422281671, −7.34356918060551036490860231734, −6.26962281518465971323726495985, −5.62466299768709886692630944237, −4.20335781028476939513820743173, −2.51163120180652293819684650484, −1.98886684377923195201891648409, 0.07196316093923038071878373447, 2.30325169668084789265606936093, 3.27856673010086445292058367399, 4.82515269264195705689331152684, 5.18273668010576577998408159295, 6.35393141270334794741931793693, 7.62632073550479476953037661639, 8.839696060426892507883139716249, 9.082816921815805226977671079469, 10.29280970708397558269327904206

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