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Properties

Label 2-496-31.25-c1-0-1
Degree $2$
Conductor $496$
Sign $0.654 - 0.755i$
Analytic cond. $3.96057$
Root an. cond. $1.99012$
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  + (−1.20 − 2.09i)3-s + (−0.5 + 0.866i)5-s + (1.20 + 2.09i)7-s + (−1.41 + 2.44i)9-s + (−2.62 + 4.54i)11-s + (−0.914 + 1.58i)13-s + 2.41·15-s + (0.0857 + 0.148i)17-s + (0.792 + 1.37i)19-s + (2.91 − 5.04i)21-s + 4·23-s + (2 + 3.46i)25-s − 0.414·27-s − 1.17·29-s + (5 − 2.44i)31-s + ⋯
L(s)  = 1  + (−0.696 − 1.20i)3-s + (−0.223 + 0.387i)5-s + (0.456 + 0.790i)7-s + (−0.471 + 0.816i)9-s + (−0.790 + 1.36i)11-s + (−0.253 + 0.439i)13-s + 0.623·15-s + (0.0208 + 0.0360i)17-s + (0.181 + 0.315i)19-s + (0.635 − 1.10i)21-s + 0.834·23-s + (0.400 + 0.692i)25-s − 0.0797·27-s − 0.217·29-s + (0.898 − 0.439i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(496\)    =    \(2^{4} \cdot 31\)
Sign: $0.654 - 0.755i$
Analytic conductor: \(3.96057\)
Root analytic conductor: \(1.99012\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{496} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 496,\ (\ :1/2),\ 0.654 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.795846 + 0.363383i\)
\(L(\frac12)\) \(\approx\) \(0.795846 + 0.363383i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 + (-5 + 2.44i)T \)
good3 \( 1 + (1.20 + 2.09i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.20 - 2.09i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.62 - 4.54i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.914 - 1.58i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.0857 - 0.148i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.792 - 1.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.74 - 8.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.44 - 7.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.65T + 47T^{2} \)
53 \( 1 + (0.0857 - 0.148i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.03 + 8.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 + (2.62 - 4.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.03 - 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.91 - 3.31i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.62 + 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.03 + 3.52i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−11.45759303415328191938257483280, −10.30686720814161083150011621932, −9.289709440597711657399284197753, −8.028023422242328128515595839629, −7.34484767424609501207374221864, −6.61332550558394286227459965530, −5.54278716848016794168472377723, −4.63420696127861785968623539996, −2.71297183493890040614677811196, −1.59839892114738659353019812549, 0.59392858944942768220851744100, 3.08939607480667841610100775596, 4.25003839727475612230267793062, 5.05540145391330960804373551304, 5.79287335707020179107481619859, 7.21690899723323282164255917170, 8.270295222001022395504521583928, 9.078986740906481524821673769038, 10.39859642377620509032758356773, 10.59334041854467119564665681356

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