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Properties

Label 2-588-147.122-c1-0-15
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: $\chi_{588} (269, \cdot )$
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

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

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