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Properties

Label 2-494-13.12-c1-0-1
Degree $2$
Conductor $494$
Sign $-0.886 - 0.462i$
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  + i·2-s − 3.29·3-s − 4-s − 3.53i·5-s − 3.29i·6-s + 0.150i·7-s − i·8-s + 7.87·9-s + 3.53·10-s + 2.38i·11-s + 3.29·12-s + (−3.19 − 1.66i)13-s − 0.150·14-s + 11.6i·15-s + 16-s − 4.64·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.90·3-s − 0.5·4-s − 1.58i·5-s − 1.34i·6-s + 0.0569i·7-s − 0.353i·8-s + 2.62·9-s + 1.11·10-s + 0.720i·11-s + 0.951·12-s + (−0.886 − 0.462i)13-s − 0.0402·14-s + 3.01i·15-s + 0.250·16-s − 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(494\)    =    \(2 \cdot 13 \cdot 19\)
Sign: $-0.886 - 0.462i$
Analytic conductor: \(3.94460\)
Root analytic conductor: \(1.98610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{494} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 494,\ (\ :1/2),\ -0.886 - 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0434408 + 0.177378i\)
\(L(\frac12)\) \(\approx\) \(0.0434408 + 0.177378i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
13 \( 1 + (3.19 + 1.66i)T \)
19 \( 1 - iT \)
good3 \( 1 + 3.29T + 3T^{2} \)
5 \( 1 + 3.53iT - 5T^{2} \)
7 \( 1 - 0.150iT - 7T^{2} \)
11 \( 1 - 2.38iT - 11T^{2} \)
17 \( 1 + 4.64T + 17T^{2} \)
23 \( 1 - 5.53T + 23T^{2} \)
29 \( 1 + 4.48T + 29T^{2} \)
31 \( 1 - 5.48iT - 31T^{2} \)
37 \( 1 - 10.9iT - 37T^{2} \)
41 \( 1 + 1.62iT - 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 - 7.02iT - 47T^{2} \)
53 \( 1 - 2.39T + 53T^{2} \)
59 \( 1 - 2.86iT - 59T^{2} \)
61 \( 1 + 4.18T + 61T^{2} \)
67 \( 1 - 9.26iT - 67T^{2} \)
71 \( 1 - 3.13iT - 71T^{2} \)
73 \( 1 - 2.07iT - 73T^{2} \)
79 \( 1 + 5.29T + 79T^{2} \)
83 \( 1 + 12.0iT - 83T^{2} \)
89 \( 1 - 18.8iT - 89T^{2} \)
97 \( 1 + 12.5iT - 97T^{2} \)
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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

−11.52846320353091010596924907398, −10.38136300067526223061046522817, −9.603322693615099462959917093572, −8.647004608131408594541725195891, −7.39518626866594019547997041554, −6.64201422357003663612053851850, −5.51305725212835124091974133335, −4.92710760482539626487631289438, −4.41651577322211365745598475956, −1.30699026208275855611314810491, 0.15204461810115118913418335018, 2.20911856929293418483376036489, 3.73573119072352204685331209770, 4.88105895921258045687571330133, 5.90901154933407114498492165784, 6.78382230259616776550097446784, 7.37734270455438115791806787456, 9.237610565691154909188826196136, 10.18635180165470714404246340513, 10.90485236440759549076805156850

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