↑

Properties

Label 2-588-147.38-c1-0-4
Degree $2$
Conductor $588$
Sign $-0.889 - 0.456i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 1.71i)3-s + (−0.288 + 2.63i)7-s + (−2.86 + 0.884i)9-s + (−6.96 + 1.59i)13-s + (2.51 − 1.45i)19-s + (−4.57 + 0.185i)21-s + (3.66 + 3.40i)25-s + (−2.25 − 4.68i)27-s + (−2.78 − 1.60i)31-s + (0.909 + 12.1i)37-s + (−4.52 − 11.5i)39-s + (−2.85 + 3.58i)43-s + (−6.83 − 1.51i)49-s + (3.13 + 3.92i)57-s + (−0.167 + 0.0125i)61-s + ⋯
L(s)  = 1  + (0.149 + 0.988i)3-s + (−0.108 + 0.994i)7-s + (−0.955 + 0.294i)9-s + (−1.93 + 0.441i)13-s + (0.576 − 0.332i)19-s + (−0.999 + 0.0405i)21-s + (0.733 + 0.680i)25-s + (−0.433 − 0.900i)27-s + (−0.499 − 0.288i)31-s + (0.149 + 1.99i)37-s + (−0.724 − 1.84i)39-s + (−0.435 + 0.546i)43-s + (−0.976 − 0.216i)49-s + (0.415 + 0.520i)57-s + (−0.0214 + 0.00160i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.889 - 0.456i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.889 - 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.237458 + 0.981805i\)
\(L(\frac12)\) \(\approx\) \(0.237458 + 0.981805i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.258 - 1.71i)T \)
7 \( 1 + (0.288 - 2.63i)T \)
good5 \( 1 + (-3.66 - 3.40i)T^{2} \)
11 \( 1 + (-9.08 - 6.19i)T^{2} \)
13 \( 1 + (6.96 - 1.59i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (-2.51 + 1.45i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-8.40 - 21.4i)T^{2} \)
29 \( 1 + (-18.0 - 22.6i)T^{2} \)
31 \( 1 + (2.78 + 1.60i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.909 - 12.1i)T + (-36.5 + 5.51i)T^{2} \)
41 \( 1 + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (2.85 - 3.58i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (52.4 + 7.89i)T^{2} \)
59 \( 1 + (-43.2 + 40.1i)T^{2} \)
61 \( 1 + (0.167 - 0.0125i)T + (60.3 - 9.09i)T^{2} \)
67 \( 1 + (-4.53 + 7.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-3.95 + 4.26i)T + (-5.45 - 72.7i)T^{2} \)
79 \( 1 + (-8.78 - 15.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (73.5 - 50.1i)T^{2} \)
97 \( 1 - 16.7iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.08745057912244814218047578247, −9.862587740005736336728697197145, −9.526780450455024281759164196933, −8.657373525567153285838976781893, −7.63520928804749668387567448291, −6.47788502536058535719493292067, −5.19809791390805625236222810798, −4.78917585856789851509840003412, −3.27373279188164110938561534319, −2.35405315537534965170532394021, 0.52805753287209597589104280182, 2.17184770714984148648077317148, 3.35294808595322940210372375943, 4.75796245920807987303649252150, 5.82719192470715108860516002125, 7.18891571602347243105672397828, 7.29428082811124303895310699523, 8.368814339827900782377299890865, 9.496917146311782244108075096382, 10.30552832688666513930765532676

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