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Properties

Label 2-538-269.187-c2-0-38
Degree $2$
Conductor $538$
Sign $-0.158 - 0.987i$
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 + (3.92 + 3.92i)3-s + 2i·4-s + 6.20·5-s + 7.85i·6-s + (5.44 − 5.44i)7-s + (−2 + 2i)8-s + 21.8i·9-s + (6.20 + 6.20i)10-s − 8.07i·11-s + (−7.85 + 7.85i)12-s + 4.08i·13-s + 10.8·14-s + (24.3 + 24.3i)15-s − 4·16-s + (−21.9 − 21.9i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (1.30 + 1.30i)3-s + 0.5i·4-s + 1.24·5-s + 1.30i·6-s + (0.777 − 0.777i)7-s + (−0.250 + 0.250i)8-s + 2.43i·9-s + (0.620 + 0.620i)10-s − 0.734i·11-s + (−0.654 + 0.654i)12-s + 0.313i·13-s + 0.777·14-s + (1.62 + 1.62i)15-s − 0.250·16-s + (−1.28 − 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $-0.158 - 0.987i$
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: \((2,\ 538,\ (\ :1),\ -0.158 - 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.474728634\)
\(L(\frac12)\) \(\approx\) \(4.474728634\)
\(L(2)\) 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 \( 1 + (-219. + 155. i)T \)
good3 \( 1 + (-3.92 - 3.92i)T + 9iT^{2} \)
5 \( 1 - 6.20T + 25T^{2} \)
7 \( 1 + (-5.44 + 5.44i)T - 49iT^{2} \)
11 \( 1 + 8.07iT - 121T^{2} \)
13 \( 1 - 4.08iT - 169T^{2} \)
17 \( 1 + (21.9 + 21.9i)T + 289iT^{2} \)
19 \( 1 + (-1.54 + 1.54i)T - 361iT^{2} \)
23 \( 1 + 6.49T + 529T^{2} \)
29 \( 1 + (18.6 - 18.6i)T - 841iT^{2} \)
31 \( 1 + (-40.4 + 40.4i)T - 961iT^{2} \)
37 \( 1 + 18.5T + 1.36e3T^{2} \)
41 \( 1 - 50.3T + 1.68e3T^{2} \)
43 \( 1 + 3.15iT - 1.84e3T^{2} \)
47 \( 1 + 57.6T + 2.20e3T^{2} \)
53 \( 1 + 92.7T + 2.80e3T^{2} \)
59 \( 1 + (-4.96 + 4.96i)T - 3.48e3iT^{2} \)
61 \( 1 + 92.6T + 3.72e3T^{2} \)
67 \( 1 + 40.9T + 4.48e3T^{2} \)
71 \( 1 + (-14.9 + 14.9i)T - 5.04e3iT^{2} \)
73 \( 1 - 67.1iT - 5.32e3T^{2} \)
79 \( 1 - 1.04iT - 6.24e3T^{2} \)
83 \( 1 + (6.35 - 6.35i)T - 6.88e3iT^{2} \)
89 \( 1 + 90.8iT - 7.92e3T^{2} \)
97 \( 1 - 67.9iT - 9.40e3T^{2} \)
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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

−10.75702762676646302773687805457, −9.681299009377567610413864095882, −9.198059132439467437234542025153, −8.298709602191540942739234654662, −7.41638251364091728696341644991, −6.10731722056478345846001196128, −4.89002142515342843528902756634, −4.37896271546036590483429669160, −3.13186790328140371451554518631, −2.09924034020208781328049120843, 1.71150291780460135430408325903, 1.95875447257641477104273980864, 3.02196835455647660755422206606, 4.56920181149566263936928302112, 5.94806488475430084692775550182, 6.56268004164251083961197512494, 7.81791055669018272354311447932, 8.638552541521305230949244266765, 9.338061787153593100237499483262, 10.27021369681386905590471391949

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