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Properties

Label 2-494-247.68-c1-0-9
Degree $2$
Conductor $494$
Sign $0.748 - 0.663i$
Analytic cond. $3.94460$
Root an. cond. $1.98610$
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  + 2-s + 2.67·3-s + 4-s + (−1.23 + 2.13i)5-s + 2.67·6-s + (−2.14 + 3.72i)7-s + 8-s + 4.17·9-s + (−1.23 + 2.13i)10-s + (−0.741 − 1.28i)11-s + 2.67·12-s + (2.46 − 2.62i)13-s + (−2.14 + 3.72i)14-s + (−3.29 + 5.70i)15-s + 16-s + (−3.03 − 5.26i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.54·3-s + 0.5·4-s + (−0.550 + 0.953i)5-s + 1.09·6-s + (−0.812 + 1.40i)7-s + 0.353·8-s + 1.39·9-s + (−0.389 + 0.674i)10-s + (−0.223 − 0.387i)11-s + 0.773·12-s + (0.684 − 0.728i)13-s + (−0.574 + 0.994i)14-s + (−0.851 + 1.47i)15-s + 0.250·16-s + (−0.737 − 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(494\)    =    \(2 \cdot 13 \cdot 19\)
Sign: $0.748 - 0.663i$
Analytic conductor: \(3.94460\)
Root analytic conductor: \(1.98610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{494} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 494,\ (\ :1/2),\ 0.748 - 0.663i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.79411 + 1.05953i\)
\(L(\frac12)\) \(\approx\) \(2.79411 + 1.05953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
13 \( 1 + (-2.46 + 2.62i)T \)
19 \( 1 + (-3.87 + 1.99i)T \)
good3 \( 1 - 2.67T + 3T^{2} \)
5 \( 1 + (1.23 - 2.13i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.14 - 3.72i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.741 + 1.28i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.03 + 5.26i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.53 - 4.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.90 + 8.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.64T + 31T^{2} \)
37 \( 1 + (1.04 + 1.81i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.10T + 41T^{2} \)
43 \( 1 + (1.64 + 2.84i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.11 - 1.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.66 - 2.88i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.78 + 4.82i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 1.68T + 61T^{2} \)
67 \( 1 + (-3.12 + 5.41i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.18 - 7.24i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (8.24 - 14.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.83 - 3.17i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + (0.0215 - 0.0373i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.28 - 16.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.30181066702256275449400571601, −9.987959860603852568241033941250, −9.213057214802927664284643697047, −8.306261921614485181095977103646, −7.47790415286998029437035844651, −6.49239133254996044974478757312, −5.42449928936901211938798098164, −3.81351884886817614628423708535, −2.88752545169682151493715302193, −2.62592538667262379332331674019, 1.51611158421144071739880749631, 3.16543338677802272530103332075, 3.99939096937989874098471127052, 4.56163075311170218620277570688, 6.41186842164244180363383000787, 7.26735177784177588269020257836, 8.232805991081405261580855240293, 8.819077421546105808776068878717, 9.965789441111556273780661623297, 10.67837386251857701738686318971

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