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Properties

Label 2-506-11.3-c1-0-1
Degree $2$
Conductor $506$
Sign $-0.164 - 0.986i$
Analytic cond. $4.04043$
Root an. cond. $2.01008$
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  + (−0.809 + 0.587i)2-s + (−0.916 − 2.82i)3-s + (0.309 − 0.951i)4-s + (−0.174 − 0.126i)5-s + (2.40 + 1.74i)6-s + (−0.406 + 1.25i)7-s + (0.309 + 0.951i)8-s + (−4.69 + 3.40i)9-s + 0.215·10-s + (−3.29 − 0.414i)11-s − 2.96·12-s + (−3.47 + 2.52i)13-s + (−0.406 − 1.25i)14-s + (−0.197 + 0.607i)15-s + (−0.809 − 0.587i)16-s + (4.03 + 2.93i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.529 − 1.62i)3-s + (0.154 − 0.475i)4-s + (−0.0779 − 0.0566i)5-s + (0.979 + 0.711i)6-s + (−0.153 + 0.472i)7-s + (0.109 + 0.336i)8-s + (−1.56 + 1.13i)9-s + 0.0681·10-s + (−0.992 − 0.125i)11-s − 0.856·12-s + (−0.964 + 0.700i)13-s + (−0.108 − 0.334i)14-s + (−0.0510 + 0.156i)15-s + (−0.202 − 0.146i)16-s + (0.978 + 0.711i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(506\)    =    \(2 \cdot 11 \cdot 23\)
Sign: $-0.164 - 0.986i$
Analytic conductor: \(4.04043\)
Root analytic conductor: \(2.01008\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{506} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 506,\ (\ :1/2),\ -0.164 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148500 + 0.175266i\)
\(L(\frac12)\) \(\approx\) \(0.148500 + 0.175266i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (3.29 + 0.414i)T \)
23 \( 1 - T \)
good3 \( 1 + (0.916 + 2.82i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.174 + 0.126i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.406 - 1.25i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.47 - 2.52i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.03 - 2.93i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.117 - 0.361i)T + (-15.3 + 11.1i)T^{2} \)
29 \( 1 + (2.19 - 6.74i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.203 - 0.148i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.330 + 1.01i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.62 - 11.1i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.37T + 43T^{2} \)
47 \( 1 + (1.98 + 6.10i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.20 - 6.69i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.80 - 8.62i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (9.13 + 6.63i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 7.04T + 67T^{2} \)
71 \( 1 + (4.78 + 3.47i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.80 + 8.62i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.70 + 3.41i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.63 - 4.09i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + (2.30 - 1.67i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.23774542532143514493271973085, −10.31106547325138956419843989702, −9.201030747038674867505841713706, −8.050582300783659269381638628559, −7.65081565784768074477723672695, −6.66447047587468100856257582492, −5.91444831666823939821873287427, −5.00524980535616619787252429919, −2.71785664198358818418695874957, −1.53179416408083377590985328095, 0.17310177694264754201038236372, 2.80026338517802130999255526139, 3.77283611549320392580230218537, 4.93620803189193885559050689221, 5.62979328560650411113328729304, 7.26676315024550540884431464612, 8.077349405417420270526386513907, 9.476768514848862629015186931679, 9.798521940914690533412189694475, 10.54289345582093951768529532187

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