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Properties

Label 2-588-196.187-c1-0-42
Degree $2$
Conductor $588$
Sign $0.981 + 0.190i$
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  + (−0.147 + 1.40i)2-s + (0.988 + 0.149i)3-s + (−1.95 − 0.414i)4-s + (−0.849 + 0.333i)5-s + (−0.355 + 1.36i)6-s + (0.217 − 2.63i)7-s + (0.870 − 2.69i)8-s + (0.955 + 0.294i)9-s + (−0.343 − 1.24i)10-s + (−1.69 − 5.50i)11-s + (−1.87 − 0.701i)12-s + (1.31 + 0.299i)13-s + (3.67 + 0.693i)14-s + (−0.890 + 0.203i)15-s + (3.65 + 1.62i)16-s + (2.98 − 4.38i)17-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)2-s + (0.570 + 0.0860i)3-s + (−0.978 − 0.207i)4-s + (−0.380 + 0.149i)5-s + (−0.145 + 0.558i)6-s + (0.0821 − 0.996i)7-s + (0.307 − 0.951i)8-s + (0.318 + 0.0982i)9-s + (−0.108 − 0.393i)10-s + (−0.511 − 1.65i)11-s + (−0.540 − 0.202i)12-s + (0.364 + 0.0831i)13-s + (0.982 + 0.185i)14-s + (−0.229 + 0.0524i)15-s + (0.914 + 0.405i)16-s + (0.724 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.981 + 0.190i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.981 + 0.190i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27161 - 0.122066i\)
\(L(\frac12)\) \(\approx\) \(1.27161 - 0.122066i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.147 - 1.40i)T \)
3 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (-0.217 + 2.63i)T \)
good5 \( 1 + (0.849 - 0.333i)T + (3.66 - 3.40i)T^{2} \)
11 \( 1 + (1.69 + 5.50i)T + (-9.08 + 6.19i)T^{2} \)
13 \( 1 + (-1.31 - 0.299i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (-2.98 + 4.38i)T + (-6.21 - 15.8i)T^{2} \)
19 \( 1 + (0.650 - 1.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.73 + 4.01i)T + (-8.40 + 21.4i)T^{2} \)
29 \( 1 + (-4.16 + 2.00i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-3.03 - 5.25i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0836 + 1.11i)T + (-36.5 - 5.51i)T^{2} \)
41 \( 1 + (-0.134 - 0.107i)T + (9.12 + 39.9i)T^{2} \)
43 \( 1 + (-3.11 + 2.48i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-0.0410 - 0.0380i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (0.0604 + 0.807i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (-5.54 + 14.1i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (4.38 + 0.328i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (-0.321 + 0.185i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.19 - 12.8i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (-3.61 - 3.89i)T + (-5.45 + 72.7i)T^{2} \)
79 \( 1 + (-11.2 - 6.50i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.172 + 0.755i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (0.528 - 1.71i)T + (-73.5 - 50.1i)T^{2} \)
97 \( 1 + 15.1iT - 97T^{2} \)
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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

−10.45195246924379631217190244466, −9.697209833619604539250443146116, −8.513559624633367626879759608493, −8.080806780727701008609064230246, −7.23473254838409586165214408727, −6.28063478804337531646642342613, −5.18699937831193051059691144141, −4.02653744432210395580423309770, −3.18867133743984384481285703891, −0.74218446727105086842181859976, 1.72032692428862383633739718830, 2.65905353195601092987435140654, 3.91476089586001326955120368437, 4.81244653185732340686406334759, 6.00343375395295934550859012772, 7.65857834978428916695613827838, 8.157668943776900249067796807324, 9.120178894066965651177663242058, 9.877192811692206695498935253152, 10.56881690451914353426972811701

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