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Properties

Label 2-588-21.2-c2-0-11
Degree $2$
Conductor $588$
Sign $-0.0751 - 0.997i$
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  + (2.29 + 1.93i)3-s + (8.17 + 4.71i)5-s + (1.49 + 8.87i)9-s + (−13.4 + 7.76i)11-s + 6.22·13-s + (9.58 + 26.6i)15-s + (2.89 − 1.67i)17-s + (−4.46 + 7.73i)19-s + (−16.3 − 9.43i)23-s + (32.0 + 55.4i)25-s + (−13.7 + 23.2i)27-s − 41.0i·29-s + (2.35 + 4.07i)31-s + (−45.8 − 8.26i)33-s + (21.9 − 37.9i)37-s + ⋯
L(s)  = 1  + (0.763 + 0.645i)3-s + (1.63 + 0.943i)5-s + (0.166 + 0.986i)9-s + (−1.22 + 0.705i)11-s + 0.479·13-s + (0.638 + 1.77i)15-s + (0.170 − 0.0983i)17-s + (−0.235 + 0.407i)19-s + (−0.710 − 0.410i)23-s + (1.28 + 2.21i)25-s + (−0.509 + 0.860i)27-s − 1.41i·29-s + (0.0759 + 0.131i)31-s + (−1.38 − 0.250i)33-s + (0.592 − 1.02i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0751 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0751 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.0751 - 0.997i$
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: \((2,\ 588,\ (\ :1),\ -0.0751 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.902480566\)
\(L(\frac12)\) \(\approx\) \(2.902480566\)
\(L(2)\) 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 \( 1 + (-2.29 - 1.93i)T \)
7 \( 1 \)
good5 \( 1 + (-8.17 - 4.71i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (13.4 - 7.76i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 6.22T + 169T^{2} \)
17 \( 1 + (-2.89 + 1.67i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (4.46 - 7.73i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (16.3 + 9.43i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 41.0iT - 841T^{2} \)
31 \( 1 + (-2.35 - 4.07i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-21.9 + 37.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 46.5iT - 1.68e3T^{2} \)
43 \( 1 - 47.8T + 1.84e3T^{2} \)
47 \( 1 + (21.1 + 12.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (1.86 - 1.07i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-11.0 + 6.38i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-9.94 + 17.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-40.4 - 70.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 65.4iT - 5.04e3T^{2} \)
73 \( 1 + (-50.2 - 87.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-10.7 + 18.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 49.3iT - 6.88e3T^{2} \)
89 \( 1 + (-40.3 - 23.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 36.3T + 9.40e3T^{2} \)
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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

−10.33494808004688165634085582951, −10.01260615561815150252973091584, −9.227852336653245268015051134208, −8.120281351762257969786223083115, −7.21073971423815234347474157926, −6.03069610205731073267945994253, −5.29255062860565769842557738592, −3.96713299531555565639579850464, −2.60060812735298382405265575881, −2.10710544537553245211919382505, 1.01467652067527225573064632144, 2.09521749339234263746483234914, 3.13382537049576762512248775010, 4.81030952077292069451585152075, 5.79094635063493075633899860643, 6.46412843154907517320594204907, 7.85465062124779311990184709541, 8.534658315163811904414411689119, 9.289290414589848763348509903934, 10.01752916709209206410735657020

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