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Properties

Label 2-588-4.3-c2-0-30
Degree $2$
Conductor $588$
Sign $0.987 + 0.154i$
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  + (−1.51 + 1.30i)2-s − 1.73i·3-s + (0.617 − 3.95i)4-s − 7.86·5-s + (2.25 + 2.63i)6-s + (4.20 + 6.80i)8-s − 2.99·9-s + (11.9 − 10.2i)10-s + 11.2i·11-s + (−6.84 − 1.07i)12-s + 10.5·13-s + 13.6i·15-s + (−15.2 − 4.88i)16-s − 25.9·17-s + (4.55 − 3.90i)18-s − 1.33i·19-s + ⋯
L(s)  = 1  + (−0.759 + 0.650i)2-s − 0.577i·3-s + (0.154 − 0.987i)4-s − 1.57·5-s + (0.375 + 0.438i)6-s + (0.525 + 0.851i)8-s − 0.333·9-s + (1.19 − 1.02i)10-s + 1.02i·11-s + (−0.570 − 0.0891i)12-s + 0.811·13-s + 0.908i·15-s + (−0.952 − 0.305i)16-s − 1.52·17-s + (0.253 − 0.216i)18-s − 0.0704i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.987 + 0.154i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ 0.987 + 0.154i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6601180104\)
\(L(\frac12)\) \(\approx\) \(0.6601180104\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.51 - 1.30i)T \)
3 \( 1 + 1.73iT \)
7 \( 1 \)
good5 \( 1 + 7.86T + 25T^{2} \)
11 \( 1 - 11.2iT - 121T^{2} \)
13 \( 1 - 10.5T + 169T^{2} \)
17 \( 1 + 25.9T + 289T^{2} \)
19 \( 1 + 1.33iT - 361T^{2} \)
23 \( 1 + 11.0iT - 529T^{2} \)
29 \( 1 + 20.4T + 841T^{2} \)
31 \( 1 - 36.9iT - 961T^{2} \)
37 \( 1 - 24.2T + 1.36e3T^{2} \)
41 \( 1 - 77.6T + 1.68e3T^{2} \)
43 \( 1 + 77.7iT - 1.84e3T^{2} \)
47 \( 1 - 0.795iT - 2.20e3T^{2} \)
53 \( 1 - 74.4T + 2.80e3T^{2} \)
59 \( 1 + 64.6iT - 3.48e3T^{2} \)
61 \( 1 - 0.601T + 3.72e3T^{2} \)
67 \( 1 + 50.3iT - 4.48e3T^{2} \)
71 \( 1 + 101. iT - 5.04e3T^{2} \)
73 \( 1 - 100.T + 5.32e3T^{2} \)
79 \( 1 + 33.4iT - 6.24e3T^{2} \)
83 \( 1 - 58.7iT - 6.88e3T^{2} \)
89 \( 1 - 53.9T + 7.92e3T^{2} \)
97 \( 1 - 39.9T + 9.40e3T^{2} \)
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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

−10.64364246180378540130262983003, −9.204504340033739082386587966740, −8.575305137757537181119905594409, −7.72498433305976103357077410853, −7.09833062473584689345346654172, −6.35772550094361842635414739122, −4.92625436684560378114361124572, −3.93826620749073575149032393245, −2.15750973753064238953103224210, −0.55411955961700288276258528276, 0.70181187163024849315675850922, 2.72365305391996870190630771995, 3.84976665838829113593186704369, 4.28406641834481518690662221103, 6.05015718894260653958933566043, 7.31485919633325283481206192024, 8.142540072953692819275359062624, 8.747240702452115947036795989269, 9.547162913846196928085335648442, 10.88326205784258482597165292548

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