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Properties

Label 2-588-21.20-c3-0-13
Degree $2$
Conductor $588$
Sign $0.884 + 0.465i$
Analytic cond. $34.6931$
Root an. cond. $5.89008$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.20 − 4.08i)3-s − 7.66·5-s + (−6.45 + 26.2i)9-s − 1.31i·11-s − 23.9i·13-s + (24.5 + 31.3i)15-s − 59.0·17-s + 28.2i·19-s + 75.8i·23-s − 66.2·25-s + (127. − 57.6i)27-s + 302. i·29-s − 93.4i·31-s + (−5.37 + 4.21i)33-s + 266.·37-s + ⋯
L(s)  = 1  + (−0.616 − 0.787i)3-s − 0.685·5-s + (−0.239 + 0.970i)9-s − 0.0360i·11-s − 0.511i·13-s + (0.422 + 0.539i)15-s − 0.843·17-s + 0.341i·19-s + 0.688i·23-s − 0.529·25-s + (0.911 − 0.410i)27-s + 1.93i·29-s − 0.541i·31-s + (−0.0283 + 0.0222i)33-s + 1.18·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.884 + 0.465i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ 0.884 + 0.465i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.022451959\)
\(L(\frac12)\) \(\approx\) \(1.022451959\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.20 + 4.08i)T \)
7 \( 1 \)
good5 \( 1 + 7.66T + 125T^{2} \)
11 \( 1 + 1.31iT - 1.33e3T^{2} \)
13 \( 1 + 23.9iT - 2.19e3T^{2} \)
17 \( 1 + 59.0T + 4.91e3T^{2} \)
19 \( 1 - 28.2iT - 6.85e3T^{2} \)
23 \( 1 - 75.8iT - 1.21e4T^{2} \)
29 \( 1 - 302. iT - 2.43e4T^{2} \)
31 \( 1 + 93.4iT - 2.97e4T^{2} \)
37 \( 1 - 266.T + 5.06e4T^{2} \)
41 \( 1 - 142.T + 6.89e4T^{2} \)
43 \( 1 - 284.T + 7.95e4T^{2} \)
47 \( 1 + 209.T + 1.03e5T^{2} \)
53 \( 1 + 629. iT - 1.48e5T^{2} \)
59 \( 1 - 730.T + 2.05e5T^{2} \)
61 \( 1 + 544. iT - 2.26e5T^{2} \)
67 \( 1 + 481.T + 3.00e5T^{2} \)
71 \( 1 + 46.5iT - 3.57e5T^{2} \)
73 \( 1 + 963. iT - 3.89e5T^{2} \)
79 \( 1 - 1.26e3T + 4.93e5T^{2} \)
83 \( 1 + 841.T + 5.71e5T^{2} \)
89 \( 1 - 1.28e3T + 7.04e5T^{2} \)
97 \( 1 - 60.2iT - 9.12e5T^{2} \)
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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

−10.49650597926661348502779006491, −9.287376892926673024990135681159, −8.176468773771974667705589087757, −7.56835399719024151020546232953, −6.66664220378027383755721728612, −5.71693974829907138094564479721, −4.73156028805143406900923059499, −3.46706730386036268458665580368, −2.03585572416187970954216501861, −0.63435468159073092316541261024, 0.59060022597831962302712117832, 2.57523899864163204784377150726, 4.08146587529253659853662273479, 4.45512344205687485719036468791, 5.78226165267970039340950631861, 6.61391818057453384749773493221, 7.70260921575023103569380082281, 8.757773629600104900184282191336, 9.536228090568477976476994964512, 10.42965155197242883627017923426

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