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Properties

Label 2-588-21.11-c2-0-12
Degree $2$
Conductor $588$
Sign $0.577 + 0.816i$
Analytic cond. $16.0218$
Root an. cond. $4.00272$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.531 − 2.95i)3-s + (−8.17 + 4.71i)5-s + (−8.43 − 3.14i)9-s + (13.4 + 7.76i)11-s + 6.22·13-s + (9.58 + 26.6i)15-s + (−2.89 − 1.67i)17-s + (−4.46 − 7.73i)19-s + (16.3 − 9.43i)23-s + (32.0 − 55.4i)25-s + (−13.7 + 23.2i)27-s − 41.0i·29-s + (2.35 − 4.07i)31-s + (30.0 − 35.5i)33-s + (21.9 + 37.9i)37-s + ⋯
L(s)  = 1  + (0.177 − 0.984i)3-s + (−1.63 + 0.943i)5-s + (−0.937 − 0.349i)9-s + (1.22 + 0.705i)11-s + 0.479·13-s + (0.638 + 1.77i)15-s + (−0.170 − 0.0983i)17-s + (−0.235 − 0.407i)19-s + (0.710 − 0.410i)23-s + (1.28 − 2.21i)25-s + (−0.509 + 0.860i)27-s − 1.41i·29-s + (0.0759 − 0.131i)31-s + (0.911 − 1.07i)33-s + (0.592 + 1.02i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.309443253\)
\(L(\frac12)\) \(\approx\) \(1.309443253\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.531 + 2.95i)T \)
7 \( 1 \)
good5 \( 1 + (8.17 - 4.71i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-13.4 - 7.76i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 6.22T + 169T^{2} \)
17 \( 1 + (2.89 + 1.67i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (4.46 + 7.73i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-16.3 + 9.43i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 41.0iT - 841T^{2} \)
31 \( 1 + (-2.35 + 4.07i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-21.9 - 37.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 46.5iT - 1.68e3T^{2} \)
43 \( 1 - 47.8T + 1.84e3T^{2} \)
47 \( 1 + (-21.1 + 12.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-1.86 - 1.07i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (11.0 + 6.38i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-9.94 - 17.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-40.4 + 70.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 65.4iT - 5.04e3T^{2} \)
73 \( 1 + (-50.2 + 87.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-10.7 - 18.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 49.3iT - 6.88e3T^{2} \)
89 \( 1 + (40.3 - 23.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 36.3T + 9.40e3T^{2} \)
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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

−10.66624482160188687778241927768, −9.263607084204807475006128344984, −8.365248650086418092581978140080, −7.56528244550561006674705255822, −6.88400698452158762262483706779, −6.26276720337324308709438288869, −4.42650710662714963578491364305, −3.56363125962773121997205700692, −2.41398359031266179487753304950, −0.67497723818562064998195193247, 0.967880031290956788505392344780, 3.32885581581857348737818793684, 3.95345987929905941221560894801, 4.73779241784249056590551885051, 5.85952996882356652025434134427, 7.23654297532133489768084468303, 8.325267897029540407115421906120, 8.786833832931509981150237616877, 9.480257729433207131232642396237, 11.01093397125353001956150041360

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