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Properties

Label 2-494-247.178-c1-0-18
Degree $2$
Conductor $494$
Sign $0.991 + 0.133i$
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 + 2.37·3-s + 4-s + (−0.834 − 1.44i)5-s + 2.37·6-s + (1.15 + 1.99i)7-s + 8-s + 2.62·9-s + (−0.834 − 1.44i)10-s + (0.0489 − 0.0847i)11-s + 2.37·12-s + (−0.883 − 3.49i)13-s + (1.15 + 1.99i)14-s + (−1.97 − 3.42i)15-s + 16-s + (−1.26 + 2.18i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.36·3-s + 0.5·4-s + (−0.373 − 0.646i)5-s + 0.968·6-s + (0.435 + 0.754i)7-s + 0.353·8-s + 0.874·9-s + (−0.263 − 0.456i)10-s + (0.0147 − 0.0255i)11-s + 0.684·12-s + (−0.244 − 0.969i)13-s + (0.308 + 0.533i)14-s + (−0.510 − 0.884i)15-s + 0.250·16-s + (−0.305 + 0.529i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(494\)    =    \(2 \cdot 13 \cdot 19\)
Sign: $0.991 + 0.133i$
Analytic conductor: \(3.94460\)
Root analytic conductor: \(1.98610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{494} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 494,\ (\ :1/2),\ 0.991 + 0.133i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.08823 - 0.207492i\)
\(L(\frac12)\) \(\approx\) \(3.08823 - 0.207492i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.883 + 3.49i)T \)
19 \( 1 + (3.53 - 2.54i)T \)
good3 \( 1 - 2.37T + 3T^{2} \)
5 \( 1 + (0.834 + 1.44i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.15 - 1.99i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0489 + 0.0847i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.26 - 2.18i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.20 - 3.82i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.89 + 3.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.143T + 31T^{2} \)
37 \( 1 + (-2.73 + 4.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.98T + 41T^{2} \)
43 \( 1 + (2.70 - 4.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.57 - 11.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.07 + 3.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.49 - 9.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
67 \( 1 + (7.39 + 12.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.848 + 1.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.54 - 2.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.855 + 1.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.64T + 83T^{2} \)
89 \( 1 + (0.711 + 1.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.622 - 1.07i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.07120229439180990396891294183, −9.917906515563037893084467364484, −8.879469792603028595709626470582, −8.226562143434313210012832355030, −7.63601125813820770084411577559, −6.13492330887687095883600881279, −5.07720092718183980411084346218, −4.02619783331948215076714504711, −2.99027406186285780833458871421, −1.89656918046273132882401016818, 2.02395181756533360664996628112, 3.09039097971270199386032595325, 4.01299235355226239184666179922, 4.91833333111003062082346378173, 6.79818635554785735981491672622, 7.11627649408527400359508235930, 8.242864703259298987218783641033, 9.019787603453488976662434468113, 10.15911980023526861596864125666, 11.07030962102419423871326772375

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