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Properties

Label 2-538-269.187-c2-0-9
Degree $2$
Conductor $538$
Sign $-0.205 + 0.978i$
Analytic cond. $14.6594$
Root an. cond. $3.82876$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−4.16 − 4.16i)3-s + 2i·4-s − 6.39·5-s + 8.33i·6-s + (−6.86 + 6.86i)7-s + (2 − 2i)8-s + 25.7i·9-s + (6.39 + 6.39i)10-s − 13.6i·11-s + (8.33 − 8.33i)12-s + 20.6i·13-s + 13.7·14-s + (26.6 + 26.6i)15-s − 4·16-s + (−12.6 − 12.6i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−1.38 − 1.38i)3-s + 0.5i·4-s − 1.27·5-s + 1.38i·6-s + (−0.981 + 0.981i)7-s + (0.250 − 0.250i)8-s + 2.85i·9-s + (0.639 + 0.639i)10-s − 1.23i·11-s + (0.694 − 0.694i)12-s + 1.58i·13-s + 0.981·14-s + (1.77 + 1.77i)15-s − 0.250·16-s + (−0.742 − 0.742i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $-0.205 + 0.978i$
Analytic conductor: \(14.6594\)
Root analytic conductor: \(3.82876\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{538} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 538,\ (\ :1),\ -0.205 + 0.978i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1377892030\)
\(L(\frac12)\) \(\approx\) \(0.1377892030\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
269 \( 1 + (-260. + 66.7i)T \)
good3 \( 1 + (4.16 + 4.16i)T + 9iT^{2} \)
5 \( 1 + 6.39T + 25T^{2} \)
7 \( 1 + (6.86 - 6.86i)T - 49iT^{2} \)
11 \( 1 + 13.6iT - 121T^{2} \)
13 \( 1 - 20.6iT - 169T^{2} \)
17 \( 1 + (12.6 + 12.6i)T + 289iT^{2} \)
19 \( 1 + (-7.34 + 7.34i)T - 361iT^{2} \)
23 \( 1 + 24.6T + 529T^{2} \)
29 \( 1 + (33.1 - 33.1i)T - 841iT^{2} \)
31 \( 1 + (24.0 - 24.0i)T - 961iT^{2} \)
37 \( 1 - 14.3T + 1.36e3T^{2} \)
41 \( 1 + 16.7T + 1.68e3T^{2} \)
43 \( 1 - 59.3iT - 1.84e3T^{2} \)
47 \( 1 + 38.1T + 2.20e3T^{2} \)
53 \( 1 + 80.2T + 2.80e3T^{2} \)
59 \( 1 + (-30.6 + 30.6i)T - 3.48e3iT^{2} \)
61 \( 1 - 44.1T + 3.72e3T^{2} \)
67 \( 1 + 70.1T + 4.48e3T^{2} \)
71 \( 1 + (-69.2 + 69.2i)T - 5.04e3iT^{2} \)
73 \( 1 + 23.2iT - 5.32e3T^{2} \)
79 \( 1 + 85.9iT - 6.24e3T^{2} \)
83 \( 1 + (-19.2 + 19.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 99.4iT - 7.92e3T^{2} \)
97 \( 1 + 40.2iT - 9.40e3T^{2} \)
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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

−10.95587273704442995935731655943, −9.386221558361746291652771183432, −8.521288224746863164180482025976, −7.54383913668963749022686229576, −6.75412217010387425790887272805, −6.05690845717952754217966944091, −4.79326917445691444604210913837, −3.24121527300247061748122171723, −1.82968379296503967318778428120, −0.22569720990034555116882429828, 0.38024788179736110048736649373, 3.77747512255128337056463563940, 4.07749919839283340367046228778, 5.32474585405366696602306979524, 6.24937463887289109544242995701, 7.20163814617622014050878055931, 8.057982073247081560730908247949, 9.551462344974473820328897136619, 10.01619486933192417969049141811, 10.65741974097457166026194167430

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