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Properties

Label 2-588-196.27-c1-0-15
Degree $2$
Conductor $588$
Sign $0.989 + 0.146i$
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  + (−0.865 − 1.11i)2-s + (0.222 − 0.974i)3-s + (−0.502 + 1.93i)4-s + (3.66 + 0.835i)5-s + (−1.28 + 0.594i)6-s + (−0.940 + 2.47i)7-s + (2.59 − 1.11i)8-s + (−0.900 − 0.433i)9-s + (−2.23 − 4.81i)10-s + (−0.615 − 1.27i)11-s + (1.77 + 0.920i)12-s + (2.92 + 6.06i)13-s + (3.57 − 1.08i)14-s + (1.62 − 3.38i)15-s + (−3.49 − 1.94i)16-s + (−2.52 + 2.01i)17-s + ⋯
L(s)  = 1  + (−0.611 − 0.790i)2-s + (0.128 − 0.562i)3-s + (−0.251 + 0.967i)4-s + (1.63 + 0.373i)5-s + (−0.523 + 0.242i)6-s + (−0.355 + 0.934i)7-s + (0.919 − 0.393i)8-s + (−0.300 − 0.144i)9-s + (−0.706 − 1.52i)10-s + (−0.185 − 0.385i)11-s + (0.512 + 0.265i)12-s + (0.809 + 1.68i)13-s + (0.956 − 0.290i)14-s + (0.420 − 0.873i)15-s + (−0.873 − 0.486i)16-s + (−0.612 + 0.488i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.989 + 0.146i$
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: \((2,\ 588,\ (\ :1/2),\ 0.989 + 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35844 - 0.100178i\)
\(L(\frac12)\) \(\approx\) \(1.35844 - 0.100178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.865 + 1.11i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
7 \( 1 + (0.940 - 2.47i)T \)
good5 \( 1 + (-3.66 - 0.835i)T + (4.50 + 2.16i)T^{2} \)
11 \( 1 + (0.615 + 1.27i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (-2.92 - 6.06i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (2.52 - 2.01i)T + (3.78 - 16.5i)T^{2} \)
19 \( 1 + 3.53T + 19T^{2} \)
23 \( 1 + (-5.43 - 4.33i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (-3.09 - 3.88i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 1.53T + 31T^{2} \)
37 \( 1 + (6.46 + 8.10i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (4.59 + 1.04i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (-1.11 + 0.253i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (-3.04 + 1.46i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-5.08 + 6.37i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (1.42 + 6.26i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-8.39 + 6.69i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 + (-5.74 - 4.57i)T + (15.7 + 69.2i)T^{2} \)
73 \( 1 + (-3.06 + 6.36i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 - 4.18iT - 79T^{2} \)
83 \( 1 + (6.05 + 2.91i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (1.93 - 4.02i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + 8.69iT - 97T^{2} \)
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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

−10.72960766116074274031453284873, −9.662178980920474202394787679339, −8.941544416669483860992003130330, −8.601143172065154504917268244413, −6.87120052941155600364638716617, −6.42106622252054587137045366464, −5.24412505920694467537192006196, −3.52412383548665931219113822191, −2.28366446120276155731581764304, −1.71821619827739986784042703919, 0.991866946988475302584422129976, 2.67248599130888984093982877444, 4.48171154290420799612368276988, 5.34320695527377144637881766322, 6.21438495724488255530665157340, 7.00390208765508612226238802280, 8.351874615317863393974398499670, 8.896499220771810507995982365275, 10.07023835064284530850762958340, 10.18687063904195052052454221815

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