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Properties

Label 2-588-196.55-c1-0-4
Degree $2$
Conductor $588$
Sign $-0.853 - 0.520i$
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  + (−1.39 − 0.205i)2-s + (−0.900 − 0.433i)3-s + (1.91 + 0.574i)4-s + (−1.72 + 3.58i)5-s + (1.17 + 0.792i)6-s + (1.74 + 1.98i)7-s + (−2.56 − 1.19i)8-s + (0.623 + 0.781i)9-s + (3.15 − 4.66i)10-s + (0.270 + 0.215i)11-s + (−1.47 − 1.34i)12-s + (4.11 + 3.28i)13-s + (−2.03 − 3.13i)14-s + (3.11 − 2.48i)15-s + (3.34 + 2.20i)16-s + (−6.18 − 1.41i)17-s + ⋯
L(s)  = 1  + (−0.989 − 0.145i)2-s + (−0.520 − 0.250i)3-s + (0.957 + 0.287i)4-s + (−0.772 + 1.60i)5-s + (0.478 + 0.323i)6-s + (0.660 + 0.750i)7-s + (−0.906 − 0.423i)8-s + (0.207 + 0.260i)9-s + (0.996 − 1.47i)10-s + (0.0814 + 0.0649i)11-s + (−0.426 − 0.389i)12-s + (1.14 + 0.910i)13-s + (−0.544 − 0.838i)14-s + (0.803 − 0.640i)15-s + (0.835 + 0.550i)16-s + (−1.49 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.853 - 0.520i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.853 - 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128957 + 0.459484i\)
\(L(\frac12)\) \(\approx\) \(0.128957 + 0.459484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.205i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (-1.74 - 1.98i)T \)
good5 \( 1 + (1.72 - 3.58i)T + (-3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.270 - 0.215i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-4.11 - 3.28i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (6.18 + 1.41i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 + 4.78T + 19T^{2} \)
23 \( 1 + (-8.08 + 1.84i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.479 - 2.10i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 5.87T + 31T^{2} \)
37 \( 1 + (0.963 - 4.22i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (0.142 - 0.295i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (0.748 + 1.55i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.26 + 1.58i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-2.60 - 11.4i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (2.99 - 1.44i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (5.09 + 1.16i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 9.75iT - 67T^{2} \)
71 \( 1 + (13.7 - 3.14i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (4.38 - 3.49i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 6.54iT - 79T^{2} \)
83 \( 1 + (-3.48 - 4.36i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-2.73 + 2.17i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 - 0.588iT - 97T^{2} \)
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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

−10.89622948741726623606895857576, −10.73931770387956183383657819843, −9.078352765475885396258411276679, −8.561610190097539064739889294889, −7.36277085623261899695749177477, −6.76564242295340620785335626790, −6.11327985170999004940281397100, −4.35476671425483072437130418531, −2.98291499628636167327935311078, −1.88923946021614213676805725614, 0.40785909181146074369010951227, 1.50827985420041652017226779570, 3.79813160500427066982557824325, 4.76064497566706690528521266486, 5.73668338818674808350613434417, 6.94711594055351158600456845781, 7.928687668107930486045516954647, 8.680202193471670831870210863850, 9.103602513969489572117743868169, 10.55213866806585017185494139824

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