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Properties

Label 2-588-4.3-c2-0-42
Degree $2$
Conductor $588$
Sign $0.938 + 0.344i$
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  + (1.14 + 1.63i)2-s − 1.73i·3-s + (−1.37 + 3.75i)4-s − 8.62·5-s + (2.84 − 1.98i)6-s + (−7.73 + 2.04i)8-s − 2.99·9-s + (−9.87 − 14.1i)10-s + 3.28i·11-s + (6.50 + 2.38i)12-s + 18.4·13-s + 14.9i·15-s + (−12.2 − 10.3i)16-s + 4.23·17-s + (−3.43 − 4.91i)18-s − 32.7i·19-s + ⋯
L(s)  = 1  + (0.572 + 0.819i)2-s − 0.577i·3-s + (−0.344 + 0.938i)4-s − 1.72·5-s + (0.473 − 0.330i)6-s + (−0.966 + 0.255i)8-s − 0.333·9-s + (−0.987 − 1.41i)10-s + 0.298i·11-s + (0.542 + 0.198i)12-s + 1.41·13-s + 0.995i·15-s + (−0.762 − 0.646i)16-s + 0.249·17-s + (−0.190 − 0.273i)18-s − 1.72i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.938 + 0.344i$
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: \((2,\ 588,\ (\ :1),\ 0.938 + 0.344i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.329175529\)
\(L(\frac12)\) \(\approx\) \(1.329175529\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.14 - 1.63i)T \)
3 \( 1 + 1.73iT \)
7 \( 1 \)
good5 \( 1 + 8.62T + 25T^{2} \)
11 \( 1 - 3.28iT - 121T^{2} \)
13 \( 1 - 18.4T + 169T^{2} \)
17 \( 1 - 4.23T + 289T^{2} \)
19 \( 1 + 32.7iT - 361T^{2} \)
23 \( 1 - 5.56iT - 529T^{2} \)
29 \( 1 - 40.3T + 841T^{2} \)
31 \( 1 + 60.5iT - 961T^{2} \)
37 \( 1 + 17.3T + 1.36e3T^{2} \)
41 \( 1 + 7.78T + 1.68e3T^{2} \)
43 \( 1 + 12.8iT - 1.84e3T^{2} \)
47 \( 1 - 3.52iT - 2.20e3T^{2} \)
53 \( 1 - 47.5T + 2.80e3T^{2} \)
59 \( 1 - 78.4iT - 3.48e3T^{2} \)
61 \( 1 - 7.02T + 3.72e3T^{2} \)
67 \( 1 - 4.35iT - 4.48e3T^{2} \)
71 \( 1 + 127. iT - 5.04e3T^{2} \)
73 \( 1 + 104.T + 5.32e3T^{2} \)
79 \( 1 + 81.1iT - 6.24e3T^{2} \)
83 \( 1 + 59.1iT - 6.88e3T^{2} \)
89 \( 1 + 35.6T + 7.92e3T^{2} \)
97 \( 1 - 95.8T + 9.40e3T^{2} \)
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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

−10.79406555831434423618213786144, −9.025658883956443690304451236928, −8.397261676736319082140350653530, −7.61120968544871788767019671936, −6.98046213922176799434437833481, −6.05005234386496953399684826400, −4.73067633225986217809323360858, −3.92545616461508755385092055772, −2.91485276720019973426037772653, −0.51757475026108874814319719875, 1.12068658725320514487446916006, 3.21566259877656813306786115864, 3.71510552723284830619682343682, 4.55620565364951294910568894453, 5.68168883876651983920778897201, 6.80894845717528995432580343350, 8.369550451743501631242192741912, 8.548375155013943956906922095993, 10.06357689703719665558694850986, 10.70919652615719280204691312945

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