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Properties

Label 2-588-196.187-c1-0-41
Degree $2$
Conductor $588$
Sign $0.990 - 0.135i$
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  + (1.39 + 0.213i)2-s + (−0.988 − 0.149i)3-s + (1.90 + 0.596i)4-s + (3.55 − 1.39i)5-s + (−1.35 − 0.419i)6-s + (0.339 + 2.62i)7-s + (2.54 + 1.24i)8-s + (0.955 + 0.294i)9-s + (5.27 − 1.19i)10-s + (−1.15 − 3.74i)11-s + (−1.79 − 0.874i)12-s + (−1.58 − 0.362i)13-s + (−0.0853 + 3.74i)14-s + (−3.72 + 0.850i)15-s + (3.28 + 2.27i)16-s + (−1.04 + 1.53i)17-s + ⋯
L(s)  = 1  + (0.988 + 0.150i)2-s + (−0.570 − 0.0860i)3-s + (0.954 + 0.298i)4-s + (1.59 − 0.624i)5-s + (−0.551 − 0.171i)6-s + (0.128 + 0.991i)7-s + (0.898 + 0.438i)8-s + (0.318 + 0.0982i)9-s + (1.66 − 0.377i)10-s + (−0.348 − 1.13i)11-s + (−0.519 − 0.252i)12-s + (−0.440 − 0.100i)13-s + (−0.0228 + 0.999i)14-s + (−0.961 + 0.219i)15-s + (0.822 + 0.569i)16-s + (−0.253 + 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.990 - 0.135i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.990 - 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.85996 + 0.194786i\)
\(L(\frac12)\) \(\approx\) \(2.85996 + 0.194786i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.213i)T \)
3 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (-0.339 - 2.62i)T \)
good5 \( 1 + (-3.55 + 1.39i)T + (3.66 - 3.40i)T^{2} \)
11 \( 1 + (1.15 + 3.74i)T + (-9.08 + 6.19i)T^{2} \)
13 \( 1 + (1.58 + 0.362i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (1.04 - 1.53i)T + (-6.21 - 15.8i)T^{2} \)
19 \( 1 + (3.08 - 5.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.78 + 5.55i)T + (-8.40 + 21.4i)T^{2} \)
29 \( 1 + (-4.41 + 2.12i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-0.0510 - 0.0883i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.846 - 11.3i)T + (-36.5 - 5.51i)T^{2} \)
41 \( 1 + (9.06 + 7.22i)T + (9.12 + 39.9i)T^{2} \)
43 \( 1 + (-9.59 + 7.64i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-5.13 - 4.76i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (-0.106 - 1.42i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (-0.218 + 0.556i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (7.99 + 0.599i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (2.51 - 1.45i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.02 - 6.27i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (9.56 + 10.3i)T + (-5.45 + 72.7i)T^{2} \)
79 \( 1 + (4.81 + 2.78i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.767 + 3.36i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (1.87 - 6.06i)T + (-73.5 - 50.1i)T^{2} \)
97 \( 1 - 1.63iT - 97T^{2} \)
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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

−10.60471381017950941492984759203, −10.21860666399239489461775248987, −8.819807479581715555507860030732, −8.143292855637204206584658091926, −6.47447988615583932025920831806, −5.94184006945009679274407928250, −5.43909521842601091110210960623, −4.44369557704894105888377757800, −2.71644278471219468867367394678, −1.75387031165218461684854580855, 1.71182708926368066439495466214, 2.71852756326239417978664583524, 4.30465204748098421699992373565, 5.07609951919093661842609323018, 6.02824178401723302876289760238, 6.91021692918434584368204099298, 7.38299008335580578504909403443, 9.432075684555755506299291266665, 10.11750368295482090485441006673, 10.65871925277162467081127403060

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