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Properties

Label 2-588-147.47-c1-0-8
Degree $2$
Conductor $588$
Sign $0.652 - 0.757i$
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  + (1.61 + 0.632i)3-s + (1.40 + 2.24i)7-s + (2.19 + 2.04i)9-s + (−0.652 + 0.148i)13-s + (0.824 + 0.476i)19-s + (0.837 + 4.50i)21-s + (−4.77 + 1.47i)25-s + (2.25 + 4.68i)27-s + (5.48 − 3.16i)31-s + (6.32 − 4.31i)37-s + (−1.14 − 0.172i)39-s + (−0.750 + 0.941i)43-s + (−3.07 + 6.28i)49-s + (1.02 + 1.28i)57-s + (−6.93 − 10.1i)61-s + ⋯
L(s)  = 1  + (0.930 + 0.365i)3-s + (0.529 + 0.848i)7-s + (0.733 + 0.680i)9-s + (−0.181 + 0.0413i)13-s + (0.189 + 0.109i)19-s + (0.182 + 0.983i)21-s + (−0.955 + 0.294i)25-s + (0.433 + 0.900i)27-s + (0.985 − 0.568i)31-s + (1.03 − 0.709i)37-s + (−0.183 − 0.0276i)39-s + (−0.114 + 0.143i)43-s + (−0.439 + 0.898i)49-s + (0.136 + 0.170i)57-s + (−0.888 − 1.30i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.652 - 0.757i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.652 - 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90581 + 0.873227i\)
\(L(\frac12)\) \(\approx\) \(1.90581 + 0.873227i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 - 0.632i)T \)
7 \( 1 + (-1.40 - 2.24i)T \)
good5 \( 1 + (4.77 - 1.47i)T^{2} \)
11 \( 1 + (-0.822 + 10.9i)T^{2} \)
13 \( 1 + (0.652 - 0.148i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-16.8 + 2.53i)T^{2} \)
19 \( 1 + (-0.824 - 0.476i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (22.7 + 3.42i)T^{2} \)
29 \( 1 + (-18.0 - 22.6i)T^{2} \)
31 \( 1 + (-5.48 + 3.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.32 + 4.31i)T + (13.5 - 34.4i)T^{2} \)
41 \( 1 + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (0.750 - 0.941i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (38.8 + 26.4i)T^{2} \)
53 \( 1 + (-19.3 - 49.3i)T^{2} \)
59 \( 1 + (56.3 + 17.3i)T^{2} \)
61 \( 1 + (6.93 + 10.1i)T + (-22.2 + 56.7i)T^{2} \)
67 \( 1 + (7.65 + 13.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-2.63 - 8.53i)T + (-60.3 + 41.1i)T^{2} \)
79 \( 1 + (-6.53 + 11.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (6.65 + 88.7i)T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−10.74690869709191240343256284970, −9.680295462687304973109013912747, −9.160821401945210087571929849870, −8.138116312353866108529457290725, −7.62946239199404176325045994703, −6.23234922192056068658197488440, −5.12680453287217691704730944046, −4.15999013413012349864742265271, −2.91891955372534257262192106204, −1.87877513215208232386808074232, 1.23429580222491473633445683700, 2.61481452692486861748329339723, 3.83623671378397927736199661320, 4.74190470363946769478789136139, 6.23055564353373942966973598192, 7.23477879407098144430694697227, 7.894565364663595169919145239929, 8.672717738725355972764029066008, 9.732332356690334863525832299113, 10.37613710293156968067178133930

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