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Properties

Label 2-495-55.54-c0-0-0
Degree $2$
Conductor $495$
Sign $1$
Analytic cond. $0.247037$
Root an. cond. $0.497028$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 1.00·4-s − 5-s − 1.41·7-s + 1.41·10-s + 11-s + 1.41·13-s + 2.00·14-s − 0.999·16-s + 1.41·17-s − 1.00·20-s − 1.41·22-s + 25-s − 2.00·26-s − 1.41·28-s + 1.41·32-s − 2.00·34-s + 1.41·35-s + 1.41·43-s + 1.00·44-s + 1.00·49-s − 1.41·50-s + 1.41·52-s − 55-s − 1.00·64-s − 1.41·65-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.00·4-s − 5-s − 1.41·7-s + 1.41·10-s + 11-s + 1.41·13-s + 2.00·14-s − 0.999·16-s + 1.41·17-s − 1.00·20-s − 1.41·22-s + 25-s − 2.00·26-s − 1.41·28-s + 1.41·32-s − 2.00·34-s + 1.41·35-s + 1.41·43-s + 1.00·44-s + 1.00·49-s − 1.41·50-s + 1.41·52-s − 55-s − 1.00·64-s − 1.41·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(0.247037\)
Root analytic conductor: \(0.497028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (109, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3618771201\)
\(L(\frac12)\) \(\approx\) \(0.3618771201\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
11 \( 1 - T \)
good2 \( 1 + 1.41T + T^{2} \)
7 \( 1 + 1.41T + T^{2} \)
13 \( 1 - 1.41T + T^{2} \)
17 \( 1 - 1.41T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.41T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.94014141968402627480855354182, −10.11070159373015564877723650903, −9.274618600428080120582852363833, −8.647097749376825137248916547471, −7.71856887820612131247399396977, −6.87673468752881853668895829004, −5.99392564629629860761584953426, −4.09074197096736824442657158173, −3.22498069863649059073817904402, −1.06479628052752460827956950528, 1.06479628052752460827956950528, 3.22498069863649059073817904402, 4.09074197096736824442657158173, 5.99392564629629860761584953426, 6.87673468752881853668895829004, 7.71856887820612131247399396977, 8.647097749376825137248916547471, 9.274618600428080120582852363833, 10.11070159373015564877723650903, 10.94014141968402627480855354182

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