↑

Properties

Label 2-165-33.17-c1-0-6
Degree $2$
Conductor $165$
Sign $0.761 + 0.648i$
Analytic cond. $1.31753$
Root an. cond. $1.14783$
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  + (0.0403 + 0.124i)2-s + (−1.72 − 0.115i)3-s + (1.60 − 1.16i)4-s + (0.951 + 0.309i)5-s + (−0.0553 − 0.219i)6-s + (−1.50 − 2.06i)7-s + (0.420 + 0.305i)8-s + (2.97 + 0.399i)9-s + 0.130i·10-s + (1.12 − 3.11i)11-s + (−2.90 + 1.82i)12-s + (4.12 − 1.34i)13-s + (0.196 − 0.270i)14-s + (−1.60 − 0.644i)15-s + (1.20 − 3.70i)16-s + (−1.85 + 5.72i)17-s + ⋯
L(s)  = 1  + (0.0285 + 0.0877i)2-s + (−0.997 − 0.0667i)3-s + (0.802 − 0.582i)4-s + (0.425 + 0.138i)5-s + (−0.0225 − 0.0894i)6-s + (−0.568 − 0.782i)7-s + (0.148 + 0.108i)8-s + (0.991 + 0.133i)9-s + 0.0412i·10-s + (0.339 − 0.940i)11-s + (−0.839 + 0.527i)12-s + (1.14 − 0.372i)13-s + (0.0524 − 0.0721i)14-s + (−0.415 − 0.166i)15-s + (0.301 − 0.926i)16-s + (−0.450 + 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.761 + 0.648i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.761 + 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00568 - 0.370365i\)
\(L(\frac12)\) \(\approx\) \(1.00568 - 0.370365i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.115i)T \)
5 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (-1.12 + 3.11i)T \)
good2 \( 1 + (-0.0403 - 0.124i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (1.50 + 2.06i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-4.12 + 1.34i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.85 - 5.72i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.284 - 0.391i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 4.22iT - 23T^{2} \)
29 \( 1 + (8.15 - 5.92i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.89 - 8.89i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.66 - 3.38i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.66 + 2.65i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.35iT - 43T^{2} \)
47 \( 1 + (0.504 - 0.694i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-8.89 + 2.89i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.80 - 6.60i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.13 - 1.34i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 4.84T + 67T^{2} \)
71 \( 1 + (1.92 + 0.624i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.412 + 0.568i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (9.67 - 3.14i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.211 - 0.651i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 5.26iT - 89T^{2} \)
97 \( 1 + (-2.30 - 7.09i)T + (-78.4 + 57.0i)T^{2} \)
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

−12.74721838987796998818134661702, −11.46677393166523895032744417714, −10.56041246100845174991633365747, −10.35327061244850215300019043762, −8.622334918494876384894228243318, −6.93008996735569478536736864708, −6.35036234489361944214588606423, −5.47623078821564545828232930682, −3.65340271887708245165605070629, −1.33739604845850678711214135498, 2.10661654235319421791046862727, 3.96981646553223542512315608591, 5.58394370784669155972409417757, 6.50038540573387716087985508226, 7.41190663458005036598652170227, 9.107104289770765232837882209575, 9.958641611350872596100300101193, 11.41122610532162875153857406810, 11.68130377152435694511654555071, 12.79110458705376646360842844317

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