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Properties

Label 2-495-11.9-c1-0-7
Degree $2$
Conductor $495$
Sign $0.733 - 0.679i$
Analytic cond. $3.95259$
Root an. cond. $1.98811$
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  + (−0.0280 + 0.0864i)2-s + (1.61 + 1.17i)4-s + (−0.309 − 0.951i)5-s + (1.98 + 1.44i)7-s + (−0.293 + 0.213i)8-s + 0.0908·10-s + (0.242 + 3.30i)11-s + (0.0999 − 0.307i)13-s + (−0.180 + 0.131i)14-s + (1.22 + 3.75i)16-s + (−1.13 − 3.48i)17-s + (−0.437 + 0.318i)19-s + (0.615 − 1.89i)20-s + (−0.292 − 0.0719i)22-s + 4.62·23-s + ⋯
L(s)  = 1  + (−0.0198 + 0.0611i)2-s + (0.805 + 0.585i)4-s + (−0.138 − 0.425i)5-s + (0.751 + 0.545i)7-s + (−0.103 + 0.0753i)8-s + 0.0287·10-s + (0.0731 + 0.997i)11-s + (0.0277 − 0.0853i)13-s + (−0.0482 + 0.0350i)14-s + (0.305 + 0.939i)16-s + (−0.274 − 0.845i)17-s + (−0.100 + 0.0729i)19-s + (0.137 − 0.423i)20-s + (−0.0624 − 0.0153i)22-s + 0.964·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.733 - 0.679i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.733 - 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63798 + 0.642159i\)
\(L(\frac12)\) \(\approx\) \(1.63798 + 0.642159i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.242 - 3.30i)T \)
good2 \( 1 + (0.0280 - 0.0864i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-1.98 - 1.44i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.0999 + 0.307i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.13 + 3.48i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.437 - 0.318i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.62T + 23T^{2} \)
29 \( 1 + (-1.19 - 0.867i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.275 + 0.847i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.84 - 1.34i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.26 - 5.27i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 6.31T + 43T^{2} \)
47 \( 1 + (1.18 - 0.859i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.19 + 9.82i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.09 - 3.69i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.00 + 6.15i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 7.05T + 67T^{2} \)
71 \( 1 + (-2.87 - 8.86i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.01 + 3.64i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.58 + 11.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.36 + 16.5i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 4.70T + 89T^{2} \)
97 \( 1 + (-3.40 + 10.4i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.36635288594313203595718325934, −10.22255379111894300385120003269, −9.108000849059970966943950486586, −8.295890707836680061114113055042, −7.44045749265072468344995552511, −6.64153518172712953172560929238, −5.31906010603521579324899912088, −4.41262763436587384626759032871, −2.92516948163940289068134765841, −1.76445187847941210309431136232, 1.22458451103129345716019865688, 2.66183348303287342428019725151, 3.95777501599042381432570256448, 5.30566232945959415245378410448, 6.27876260592506883106014834298, 7.11351185777378657617611522840, 8.041751378404098113124199524050, 9.049574717164213083619700951192, 10.33378438185729330092308198233, 10.87667738071610412124529943461

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