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Properties

Label 2-494-13.10-c1-0-16
Degree $2$
Conductor $494$
Sign $-0.204 + 0.978i$
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  + (0.866 − 0.5i)2-s + (−0.258 − 0.447i)3-s + (0.499 − 0.866i)4-s + 1.53i·5-s + (−0.447 − 0.258i)6-s + (−3.26 − 1.88i)7-s − 0.999i·8-s + (1.36 − 2.36i)9-s + (0.766 + 1.32i)10-s + (1.19 − 0.691i)11-s − 0.516·12-s + (−1.17 − 3.41i)13-s − 3.77·14-s + (0.685 − 0.395i)15-s + (−0.5 − 0.866i)16-s + (3.16 − 5.48i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.149 − 0.258i)3-s + (0.249 − 0.433i)4-s + 0.685i·5-s + (−0.182 − 0.105i)6-s + (−1.23 − 0.712i)7-s − 0.353i·8-s + (0.455 − 0.789i)9-s + (0.242 + 0.419i)10-s + (0.361 − 0.208i)11-s − 0.149·12-s + (−0.324 − 0.945i)13-s − 1.00·14-s + (0.176 − 0.102i)15-s + (−0.125 − 0.216i)16-s + (0.768 − 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(494\)    =    \(2 \cdot 13 \cdot 19\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(3.94460\)
Root analytic conductor: \(1.98610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{494} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 494,\ (\ :1/2),\ -0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02752 - 1.26417i\)
\(L(\frac12)\) \(\approx\) \(1.02752 - 1.26417i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (1.17 + 3.41i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
good3 \( 1 + (0.258 + 0.447i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 1.53iT - 5T^{2} \)
7 \( 1 + (3.26 + 1.88i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.19 + 0.691i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.16 + 5.48i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.939 + 1.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.905 - 1.56i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.35iT - 31T^{2} \)
37 \( 1 + (2.93 - 1.69i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.58 + 3.80i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.84 - 6.65i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.24iT - 47T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 + (-2.08 - 1.20i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.80 + 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.55 - 1.47i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.34 - 4.23i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.17iT - 73T^{2} \)
79 \( 1 - 1.27T + 79T^{2} \)
83 \( 1 + 8.99iT - 83T^{2} \)
89 \( 1 + (-1.22 + 0.709i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.9 - 8.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.66106570382196091964883745640, −9.986146728768056060757637866749, −9.300992616244132835655574681515, −7.59901450443147816890288122688, −6.80512173389519401611383485673, −6.23403213715878090737552326974, −4.92429784392367111043865698522, −3.45956447548790670481595799350, −3.03314729763978088605671392336, −0.848262129637673029960605870927, 2.05492319552129238203791484803, 3.60334675134372809939817662154, 4.55814141988873203689507765168, 5.58535666120535533880998480842, 6.38615191954350657620095145032, 7.43974647555731518854139129590, 8.530660884073697633879396419624, 9.454796911351842963562333729379, 10.16015752447201419749745222433, 11.42455626950172650711473562261

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