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Properties

Label 2-538-269.191-c1-0-0
Degree $2$
Conductor $538$
Sign $0.768 - 0.639i$
Analytic cond. $4.29595$
Root an. cond. $2.07266$
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  + (−0.186 − 0.982i)2-s + (−2.11 + 0.0496i)3-s + (−0.930 + 0.366i)4-s + (−0.0895 + 0.339i)5-s + (0.443 + 2.07i)6-s + (−1.39 − 3.10i)7-s + (0.533 + 0.845i)8-s + (1.48 − 0.0698i)9-s + (0.350 + 0.0246i)10-s + (−0.185 − 0.208i)11-s + (1.95 − 0.822i)12-s + (−0.312 + 4.43i)13-s + (−2.79 + 1.95i)14-s + (0.172 − 0.723i)15-s + (0.731 − 0.681i)16-s + (−0.914 − 0.813i)17-s + ⋯
L(s)  = 1  + (−0.131 − 0.694i)2-s + (−1.22 + 0.0286i)3-s + (−0.465 + 0.183i)4-s + (−0.0400 + 0.151i)5-s + (0.181 + 0.845i)6-s + (−0.527 − 1.17i)7-s + (0.188 + 0.299i)8-s + (0.496 − 0.0232i)9-s + (0.110 + 0.00780i)10-s + (−0.0557 − 0.0627i)11-s + (0.563 − 0.237i)12-s + (−0.0867 + 1.23i)13-s + (−0.746 + 0.521i)14-s + (0.0446 − 0.186i)15-s + (0.182 − 0.170i)16-s + (−0.221 − 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $0.768 - 0.639i$
Analytic conductor: \(4.29595\)
Root analytic conductor: \(2.07266\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{538} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 538,\ (\ :1/2),\ 0.768 - 0.639i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.433825 + 0.156975i\)
\(L(\frac12)\) \(\approx\) \(0.433825 + 0.156975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.186 + 0.982i)T \)
269 \( 1 + (-12.7 + 10.2i)T \)
good3 \( 1 + (2.11 - 0.0496i)T + (2.99 - 0.140i)T^{2} \)
5 \( 1 + (0.0895 - 0.339i)T + (-4.34 - 2.46i)T^{2} \)
7 \( 1 + (1.39 + 3.10i)T + (-4.65 + 5.23i)T^{2} \)
11 \( 1 + (0.185 + 0.208i)T + (-1.28 + 10.9i)T^{2} \)
13 \( 1 + (0.312 - 4.43i)T + (-12.8 - 1.82i)T^{2} \)
17 \( 1 + (0.914 + 0.813i)T + (1.98 + 16.8i)T^{2} \)
19 \( 1 + (3.87 + 2.57i)T + (7.37 + 17.5i)T^{2} \)
23 \( 1 + (-0.0979 - 4.17i)T + (-22.9 + 1.07i)T^{2} \)
29 \( 1 + (-4.54 - 8.02i)T + (-14.8 + 24.8i)T^{2} \)
31 \( 1 + (-10.2 - 1.20i)T + (30.1 + 7.20i)T^{2} \)
37 \( 1 + (2.48 - 4.62i)T + (-20.4 - 30.8i)T^{2} \)
41 \( 1 + (6.28 + 1.19i)T + (38.1 + 15.0i)T^{2} \)
43 \( 1 + (9.07 - 5.14i)T + (22.0 - 36.8i)T^{2} \)
47 \( 1 + (-10.5 - 3.04i)T + (39.7 + 25.0i)T^{2} \)
53 \( 1 + (4.28 + 5.83i)T + (-15.9 + 50.5i)T^{2} \)
59 \( 1 + (-2.34 - 5.94i)T + (-43.1 + 40.2i)T^{2} \)
61 \( 1 + (-4.00 - 4.95i)T + (-12.7 + 59.6i)T^{2} \)
67 \( 1 + (10.3 + 4.07i)T + (49.0 + 45.6i)T^{2} \)
71 \( 1 + (-2.71 + 3.89i)T + (-24.4 - 66.6i)T^{2} \)
73 \( 1 + (-1.81 + 4.95i)T + (-55.6 - 47.2i)T^{2} \)
79 \( 1 + (1.66 + 1.28i)T + (20.1 + 76.3i)T^{2} \)
83 \( 1 + (-0.324 - 3.45i)T + (-81.5 + 15.4i)T^{2} \)
89 \( 1 + (-2.63 - 15.9i)T + (-84.2 + 28.6i)T^{2} \)
97 \( 1 + (10.8 - 13.4i)T + (-20.3 - 94.8i)T^{2} \)
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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

−10.79358081022961928242135003675, −10.45509176572175771262701002088, −9.440261426488912999162186417813, −8.422186939419905047121506564190, −6.82003462799178654335570865742, −6.68548743533151509841174334720, −5.06836239252873610206949381154, −4.34117648530948209862513722050, −3.06170629984876389566240287212, −1.17886534660792920120901462759, 0.38009632154253752742639172445, 2.68942875559965774623087431887, 4.47157285032332860450150010202, 5.42762189300108239836608904723, 6.11832954013932062395898228272, 6.68914919664496340891775579999, 8.225627187353363296339432787579, 8.660235885126521162904355305225, 10.07654434792965059114738425275, 10.48952010103019546366940683734

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