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Properties

Label 2-588-196.55-c1-0-43
Degree $2$
Conductor $588$
Sign $-0.114 + 0.993i$
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  + (−0.446 + 1.34i)2-s + (0.900 + 0.433i)3-s + (−1.60 − 1.19i)4-s + (−0.0202 + 0.0419i)5-s + (−0.984 + 1.01i)6-s + (−2.50 + 0.851i)7-s + (2.32 − 1.61i)8-s + (0.623 + 0.781i)9-s + (−0.0473 − 0.0458i)10-s + (−4.22 − 3.36i)11-s + (−0.923 − 1.77i)12-s + (−4.09 − 3.26i)13-s + (−0.0248 − 3.74i)14-s + (−0.0364 + 0.0290i)15-s + (1.13 + 3.83i)16-s + (−1.81 − 0.414i)17-s + ⋯
L(s)  = 1  + (−0.315 + 0.948i)2-s + (0.520 + 0.250i)3-s + (−0.800 − 0.598i)4-s + (−0.00904 + 0.0187i)5-s + (−0.401 + 0.414i)6-s + (−0.946 + 0.321i)7-s + (0.820 − 0.571i)8-s + (0.207 + 0.260i)9-s + (−0.0149 − 0.0145i)10-s + (−1.27 − 1.01i)11-s + (−0.266 − 0.512i)12-s + (−1.13 − 0.906i)13-s + (−0.00662 − 0.999i)14-s + (−0.00940 + 0.00750i)15-s + (0.283 + 0.959i)16-s + (−0.440 − 0.100i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.114 + 0.993i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123726 - 0.138836i\)
\(L(\frac12)\) \(\approx\) \(0.123726 - 0.138836i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.446 - 1.34i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
7 \( 1 + (2.50 - 0.851i)T \)
good5 \( 1 + (0.0202 - 0.0419i)T + (-3.11 - 3.90i)T^{2} \)
11 \( 1 + (4.22 + 3.36i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (4.09 + 3.26i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (1.81 + 0.414i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 - 3.64T + 19T^{2} \)
23 \( 1 + (7.77 - 1.77i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.460 + 2.01i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 6.23T + 31T^{2} \)
37 \( 1 + (-1.79 + 7.84i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (1.75 - 3.63i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (2.31 + 4.80i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (0.371 - 0.466i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.113 - 0.497i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (3.00 - 1.44i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-12.0 - 2.74i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 9.62iT - 67T^{2} \)
71 \( 1 + (1.82 - 0.417i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-7.50 + 5.98i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 6.41iT - 79T^{2} \)
83 \( 1 + (2.99 + 3.75i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (12.8 - 10.2i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + 6.08iT - 97T^{2} \)
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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

−10.06109637688961454439495246094, −9.571610979960522766102872598981, −8.593042333640152624272334479693, −7.80255714249056334703275297807, −7.09367799354379632481808532332, −5.71748444451650888600194840130, −5.29935761490384423964175644956, −3.74379089682038676612605882212, −2.62222072157898292029904535064, −0.10074658488350615706132226568, 2.02293473722118677248647827762, 2.85884653562806110108435587363, 4.10567021024203055931046846388, 5.04105967852501358118338187475, 6.75154338082671748724090537530, 7.53583423504534849564503919427, 8.383003043409828752693989788064, 9.629817373817579986382852894309, 9.817946331873286357329882941366, 10.71515775115859099086251134987

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