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Properties

Label 2-588-28.27-c1-0-19
Degree $2$
Conductor $588$
Sign $-0.445 + 0.895i$
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  + (−0.476 − 1.33i)2-s − 3-s + (−1.54 + 1.26i)4-s − 0.509i·5-s + (0.476 + 1.33i)6-s + (2.42 + 1.45i)8-s + 9-s + (−0.678 + 0.243i)10-s + 4.12i·11-s + (1.54 − 1.26i)12-s − 3.97i·13-s + 0.509i·15-s + (0.778 − 3.92i)16-s − 4.93i·17-s + (−0.476 − 1.33i)18-s + 5.05·19-s + ⋯
L(s)  = 1  + (−0.336 − 0.941i)2-s − 0.577·3-s + (−0.772 + 0.634i)4-s − 0.228i·5-s + (0.194 + 0.543i)6-s + (0.857 + 0.513i)8-s + 0.333·9-s + (−0.214 + 0.0768i)10-s + 1.24i·11-s + (0.446 − 0.366i)12-s − 1.10i·13-s + 0.131i·15-s + (0.194 − 0.980i)16-s − 1.19i·17-s + (−0.112 − 0.313i)18-s + 1.15·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.445 + 0.895i$
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: \((2,\ 588,\ (\ :1/2),\ -0.445 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.443172 - 0.715641i\)
\(L(\frac12)\) \(\approx\) \(0.443172 - 0.715641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.476 + 1.33i)T \)
3 \( 1 + T \)
7 \( 1 \)
good5 \( 1 + 0.509iT - 5T^{2} \)
11 \( 1 - 4.12iT - 11T^{2} \)
13 \( 1 + 3.97iT - 13T^{2} \)
17 \( 1 + 4.93iT - 17T^{2} \)
19 \( 1 - 5.05T + 19T^{2} \)
23 \( 1 + 2.91iT - 23T^{2} \)
29 \( 1 + 5.82T + 29T^{2} \)
31 \( 1 + 5.71T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 + 11.7iT - 41T^{2} \)
43 \( 1 + 9.84iT - 43T^{2} \)
47 \( 1 + 7.87T + 47T^{2} \)
53 \( 1 - 1.23T + 53T^{2} \)
59 \( 1 - 4.65T + 59T^{2} \)
61 \( 1 + 0.543iT - 61T^{2} \)
67 \( 1 + 7.02iT - 67T^{2} \)
71 \( 1 + 1.06iT - 71T^{2} \)
73 \( 1 - 0.837iT - 73T^{2} \)
79 \( 1 - 1.25iT - 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 4.68iT - 89T^{2} \)
97 \( 1 - 1.80iT - 97T^{2} \)
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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

−10.45000830781601848837441664961, −9.677234813571228659319604919915, −9.006744775134801188895637099322, −7.69857058712872942582307224884, −7.13233467321143550898500141158, −5.41957340323768218487574404402, −4.81964554301527193591799338919, −3.54470271405179054786278605304, −2.23747750896269435079692223823, −0.66396054035713982276296485623, 1.30524912665631362720422055321, 3.52640011908604988402495134500, 4.70517476670298907958340743048, 5.80599685774670232122478130479, 6.34808965506211194842589526961, 7.36398784806455325836792111749, 8.217853853778889791733306326066, 9.212881270846998161557844899518, 9.889059604228038286586480763982, 11.12290438562394619518571851678

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