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Properties

Label 2-588-49.22-c1-0-5
Degree $2$
Conductor $588$
Sign $0.994 + 0.106i$
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  + (0.222 − 0.974i)3-s + (−0.526 + 2.30i)5-s + (2.10 − 1.60i)7-s + (−0.900 − 0.433i)9-s + (−3.55 + 1.71i)11-s + (5.87 − 2.83i)13-s + (2.13 + 1.02i)15-s + (4.13 + 5.18i)17-s + 2.21·19-s + (−1.09 − 2.40i)21-s + (4.72 − 5.92i)23-s + (−0.532 − 0.256i)25-s + (−0.623 + 0.781i)27-s + (0.460 + 0.577i)29-s − 8.39·31-s + ⋯
L(s)  = 1  + (0.128 − 0.562i)3-s + (−0.235 + 1.03i)5-s + (0.794 − 0.606i)7-s + (−0.300 − 0.144i)9-s + (−1.07 + 0.516i)11-s + (1.63 − 0.785i)13-s + (0.550 + 0.264i)15-s + (1.00 + 1.25i)17-s + 0.507·19-s + (−0.239 − 0.525i)21-s + (0.986 − 1.23i)23-s + (−0.106 − 0.0512i)25-s + (−0.119 + 0.150i)27-s + (0.0854 + 0.107i)29-s − 1.50·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.994 + 0.106i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.994 + 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64185 - 0.0875121i\)
\(L(\frac12)\) \(\approx\) \(1.64185 - 0.0875121i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.222 + 0.974i)T \)
7 \( 1 + (-2.10 + 1.60i)T \)
good5 \( 1 + (0.526 - 2.30i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (3.55 - 1.71i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-5.87 + 2.83i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-4.13 - 5.18i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 - 2.21T + 19T^{2} \)
23 \( 1 + (-4.72 + 5.92i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (-0.460 - 0.577i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + 8.39T + 31T^{2} \)
37 \( 1 + (-7.05 - 8.84i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (-1.26 + 5.56i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-0.152 - 0.666i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-1.36 + 0.657i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (0.687 - 0.861i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (0.367 + 1.60i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (2.87 + 3.60i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 - 5.10T + 67T^{2} \)
71 \( 1 + (1.95 - 2.45i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (6.58 + 3.17i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 - 3.33T + 79T^{2} \)
83 \( 1 + (0.719 + 0.346i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (15.1 + 7.29i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + 15.5T + 97T^{2} \)
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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

−10.79594311669788442504194208500, −10.16747018617789833676565817539, −8.581852437836951571534076674365, −7.913312743689159102616962799079, −7.27048571484548369410813273591, −6.25954227503181774800909181921, −5.24472194968601514115005211975, −3.79481739881824181415856723258, −2.84247171689210156219611805981, −1.29496930598657351615973690845, 1.23693901274441626283865783721, 2.96873470619246545515484803103, 4.16984283626842643066360723680, 5.30462188108288120732835490707, 5.62293831803049250814413323533, 7.43693658196608863273615029984, 8.228289567848940623745649046680, 9.012072797202152514671215768510, 9.522684568708189564491322872887, 11.11165764334321215503067900836

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