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Properties

Label 2-588-28.27-c1-0-33
Degree $2$
Conductor $588$
Sign $0.998 - 0.0536i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.33 + 0.458i)2-s + 3-s + (1.57 + 1.22i)4-s − 2.45i·5-s + (1.33 + 0.458i)6-s + (1.55 + 2.36i)8-s + 9-s + (1.12 − 3.28i)10-s − 1.26i·11-s + (1.57 + 1.22i)12-s − 2.99i·13-s − 2.45i·15-s + (0.992 + 3.87i)16-s + 1.83i·17-s + (1.33 + 0.458i)18-s + 4.15·19-s + ⋯
L(s)  = 1  + (0.946 + 0.324i)2-s + 0.577·3-s + (0.789 + 0.613i)4-s − 1.09i·5-s + (0.546 + 0.187i)6-s + (0.548 + 0.836i)8-s + 0.333·9-s + (0.355 − 1.03i)10-s − 0.381i·11-s + (0.456 + 0.354i)12-s − 0.831i·13-s − 0.633i·15-s + (0.248 + 0.968i)16-s + 0.444i·17-s + (0.315 + 0.108i)18-s + 0.954·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.998 - 0.0536i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.998 - 0.0536i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.13315 + 0.0841100i\)
\(L(\frac12)\) \(\approx\) \(3.13315 + 0.0841100i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 - 0.458i)T \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + 2.45iT - 5T^{2} \)
11 \( 1 + 1.26iT - 11T^{2} \)
13 \( 1 + 2.99iT - 13T^{2} \)
17 \( 1 - 1.83iT - 17T^{2} \)
19 \( 1 - 4.15T + 19T^{2} \)
23 \( 1 - 6.73iT - 23T^{2} \)
29 \( 1 + 9.42T + 29T^{2} \)
31 \( 1 + 9.43T + 31T^{2} \)
37 \( 1 - 7.51T + 37T^{2} \)
41 \( 1 - 1.08iT - 41T^{2} \)
43 \( 1 + 6.27iT - 43T^{2} \)
47 \( 1 + 7.35T + 47T^{2} \)
53 \( 1 + 0.0716T + 53T^{2} \)
59 \( 1 + 3.36T + 59T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 + 2.80iT - 67T^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 - 8.10iT - 73T^{2} \)
79 \( 1 - 1.78iT - 79T^{2} \)
83 \( 1 + 5.33T + 83T^{2} \)
89 \( 1 + 8.57iT - 89T^{2} \)
97 \( 1 - 7.10iT - 97T^{2} \)
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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

−10.94937823700787865335639255746, −9.644464358476241053399140498440, −8.817073117736888412789263785007, −7.87715774864153886087909991279, −7.29391966874864004490658918508, −5.73782671660150148261219009969, −5.28944010667167460596130214492, −4.01478615756406289244188618418, −3.18923047777468384035385363332, −1.61599471203690492856346109821, 1.93056326903974806631155284945, 2.92885874166160792111687522660, 3.84629860290117885944706242909, 4.92956644067921718385006494132, 6.19126517922955803745003699841, 7.03044667335578690117842805033, 7.65514926478361886869573485461, 9.244190134641466834602461937620, 9.886462952958565913894355422668, 10.98009475536737467532270373270

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