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Properties

Label 2-588-196.27-c1-0-0
Degree $2$
Conductor $588$
Sign $-0.782 + 0.622i$
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  + (−1.29 + 0.577i)2-s + (−0.222 + 0.974i)3-s + (1.33 − 1.49i)4-s + (−3.29 − 0.752i)5-s + (−0.275 − 1.38i)6-s + (2.53 − 0.741i)7-s + (−0.860 + 2.69i)8-s + (−0.900 − 0.433i)9-s + (4.68 − 0.931i)10-s + (0.238 + 0.496i)11-s + (1.15 + 1.63i)12-s + (1.89 + 3.93i)13-s + (−2.85 + 2.42i)14-s + (1.46 − 3.04i)15-s + (−0.445 − 3.97i)16-s + (−0.912 + 0.728i)17-s + ⋯
L(s)  = 1  + (−0.912 + 0.408i)2-s + (−0.128 + 0.562i)3-s + (0.666 − 0.745i)4-s + (−1.47 − 0.336i)5-s + (−0.112 − 0.566i)6-s + (0.959 − 0.280i)7-s + (−0.304 + 0.952i)8-s + (−0.300 − 0.144i)9-s + (1.48 − 0.294i)10-s + (0.0720 + 0.149i)11-s + (0.333 + 0.470i)12-s + (0.525 + 1.09i)13-s + (−0.761 + 0.647i)14-s + (0.378 − 0.786i)15-s + (−0.111 − 0.993i)16-s + (−0.221 + 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.782 + 0.622i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.782 + 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00356722 - 0.0102116i\)
\(L(\frac12)\) \(\approx\) \(0.00356722 - 0.0102116i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 - 0.577i)T \)
3 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (-2.53 + 0.741i)T \)
good5 \( 1 + (3.29 + 0.752i)T + (4.50 + 2.16i)T^{2} \)
11 \( 1 + (-0.238 - 0.496i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.89 - 3.93i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (0.912 - 0.728i)T + (3.78 - 16.5i)T^{2} \)
19 \( 1 + 5.82T + 19T^{2} \)
23 \( 1 + (3.59 + 2.86i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (2.97 + 3.73i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 + (3.96 + 4.96i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (-5.88 - 1.34i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (4.65 - 1.06i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (-3.91 + 1.88i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (2.02 - 2.54i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (1.10 + 4.82i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (11.4 - 9.16i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + 6.97iT - 67T^{2} \)
71 \( 1 + (4.40 + 3.51i)T + (15.7 + 69.2i)T^{2} \)
73 \( 1 + (4.05 - 8.41i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + 4.40iT - 79T^{2} \)
83 \( 1 + (12.3 + 5.93i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (7.13 - 14.8i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + 2.32iT - 97T^{2} \)
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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

−11.08576625335876827059744393206, −10.52326126263936146521390662911, −9.163901407668119081282406409996, −8.605708052304962277111091368447, −7.85003963566246598283599603926, −7.05217219578094559543155479842, −5.87968385437985136738962609120, −4.52934578327510477479279991348, −3.97708283454921277277939914638, −1.87191546216255921235168080084, 0.008207774155288253976257076660, 1.69658298827286277436172284728, 3.14379233730032804587238266134, 4.15231041304453540172701624230, 5.73171220215289271493050520049, 7.00799991647347868657660944179, 7.73241189771093820700103171105, 8.274105042525984579364357078869, 8.984535464225847611800656871259, 10.57597881776789706813284173811

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