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Properties

Label 2-588-49.4-c1-0-1
Degree $2$
Conductor $588$
Sign $0.537 - 0.843i$
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  + (−0.733 − 0.680i)3-s + (−2.74 + 0.848i)5-s + (−0.420 − 2.61i)7-s + (0.0747 + 0.997i)9-s + (−0.344 + 4.59i)11-s + (3.74 − 1.80i)13-s + (2.59 + 1.24i)15-s + (−3.76 + 0.568i)17-s + (2.88 + 4.99i)19-s + (−1.46 + 2.20i)21-s + (7.21 + 1.08i)23-s + (2.71 − 1.84i)25-s + (0.623 − 0.781i)27-s + (4.07 + 5.10i)29-s + (−5.29 + 9.16i)31-s + ⋯
L(s)  = 1  + (−0.423 − 0.392i)3-s + (−1.22 + 0.379i)5-s + (−0.159 − 0.987i)7-s + (0.0249 + 0.332i)9-s + (−0.103 + 1.38i)11-s + (1.03 − 0.499i)13-s + (0.669 + 0.322i)15-s + (−0.914 + 0.137i)17-s + (0.662 + 1.14i)19-s + (−0.320 + 0.480i)21-s + (1.50 + 0.226i)23-s + (0.542 − 0.369i)25-s + (0.119 − 0.150i)27-s + (0.756 + 0.948i)29-s + (−0.950 + 1.64i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.537 - 0.843i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.537 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.714413 + 0.391891i\)
\(L(\frac12)\) \(\approx\) \(0.714413 + 0.391891i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.733 + 0.680i)T \)
7 \( 1 + (0.420 + 2.61i)T \)
good5 \( 1 + (2.74 - 0.848i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (0.344 - 4.59i)T + (-10.8 - 1.63i)T^{2} \)
13 \( 1 + (-3.74 + 1.80i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (3.76 - 0.568i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-2.88 - 4.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.21 - 1.08i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (-4.07 - 5.10i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (5.29 - 9.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.10 + 5.35i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (0.158 - 0.695i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-0.338 - 1.48i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (2.39 + 1.62i)T + (17.1 + 43.7i)T^{2} \)
53 \( 1 + (-2.92 - 7.46i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (-3.13 - 0.966i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (-2.12 + 5.41i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (2.43 - 4.22i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.48 - 1.85i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-3.53 + 2.40i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-5.01 - 8.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.1 + 4.88i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (0.753 + 10.0i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−10.81526240857611581764245787721, −10.34221821419066823843730064396, −8.991754826369847661037407910496, −7.892312140239676484110747382232, −7.21180835188590262094656625676, −6.68924090862122150478406985871, −5.20412642114044779286355258873, −4.14501584623586847314915847239, −3.26444175920903777443295105534, −1.29600709925010558560593096887, 0.55655531853327784475904670460, 2.86251078352241355796225642662, 3.92911124575655821941595322978, 4.92270433095901111148120565680, 5.93323908795766968858712516970, 6.84116962354522570638583458609, 8.184068533690381243341238290098, 8.769767831522366851528762049578, 9.443331996299135003749233565002, 11.06299191086530836977437423298

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