↑

Properties

Label 2-495-11.3-c1-0-19
Degree $2$
Conductor $495$
Sign $-0.631 + 0.775i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.24 − 1.62i)2-s + (1.75 − 5.40i)4-s + (−0.809 − 0.587i)5-s + (−0.703 + 2.16i)7-s + (−3.15 − 9.71i)8-s − 2.77·10-s + (0.105 − 3.31i)11-s + (0.352 − 0.256i)13-s + (1.95 + 6.00i)14-s + (−13.7 − 9.96i)16-s + (4.04 + 2.93i)17-s + (1.45 + 4.46i)19-s + (−4.59 + 3.34i)20-s + (−5.16 − 7.60i)22-s + 0.845·23-s + ⋯
L(s)  = 1  + (1.58 − 1.15i)2-s + (0.878 − 2.70i)4-s + (−0.361 − 0.262i)5-s + (−0.266 + 0.818i)7-s + (−1.11 − 3.43i)8-s − 0.876·10-s + (0.0317 − 0.999i)11-s + (0.0977 − 0.0710i)13-s + (0.521 + 1.60i)14-s + (−3.42 − 2.49i)16-s + (0.981 + 0.712i)17-s + (0.332 + 1.02i)19-s + (−1.02 + 0.747i)20-s + (−1.10 − 1.62i)22-s + 0.176·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.631 + 0.775i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.631 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31488 - 2.76788i\)
\(L(\frac12)\) \(\approx\) \(1.31488 - 2.76788i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-0.105 + 3.31i)T \)
good2 \( 1 + (-2.24 + 1.62i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.703 - 2.16i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.352 + 0.256i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.04 - 2.93i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.45 - 4.46i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.845T + 23T^{2} \)
29 \( 1 + (-0.821 + 2.52i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.77 + 2.74i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.73 - 8.42i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.32 - 4.08i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.00T + 43T^{2} \)
47 \( 1 + (0.144 + 0.445i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.76 - 6.37i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.21 - 3.74i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.39 + 1.74i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.47T + 67T^{2} \)
71 \( 1 + (9.15 + 6.65i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.60 + 8.01i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.79 + 6.38i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.78 + 3.47i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 5.89T + 89T^{2} \)
97 \( 1 + (-6.99 + 5.08i)T + (29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.90136803177402991857566043305, −10.13243923367246570723749113823, −9.179386423182071710007064111726, −7.915774365719782108780554684883, −6.15886796138985381383839815456, −5.84365426855739927301970089068, −4.70698846862941386410658304183, −3.59519868642309678566868570090, −2.85124327743109699521355329062, −1.30525306846147425815365892127, 2.76568968074508321943920098542, 3.78721814042650933438278099571, 4.65682539411112667291245501807, 5.53571653990506611764306171653, 6.90193814257753402237945955982, 7.12518042773660321890478019830, 8.000784524510425230781290100850, 9.288299542577135558078023307622, 10.67205925651296663054231602215, 11.64669999549781248515174359336

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