↑

Properties

Label 2-165-55.9-c1-0-7
Degree $2$
Conductor $165$
Sign $-0.987 + 0.154i$
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  + (−2.10 − 0.684i)2-s + (−0.587 − 0.809i)3-s + (2.34 + 1.70i)4-s + (−0.0486 − 2.23i)5-s + (0.684 + 2.10i)6-s + (0.936 − 1.28i)7-s + (−1.17 − 1.61i)8-s + (−0.309 + 0.951i)9-s + (−1.42 + 4.74i)10-s + (−2.17 − 2.50i)11-s − 2.90i·12-s + (−0.847 − 0.275i)13-s + (−2.85 + 2.07i)14-s + (−1.77 + 1.35i)15-s + (−0.427 − 1.31i)16-s + (−4.22 + 1.37i)17-s + ⋯
L(s)  = 1  + (−1.48 − 0.483i)2-s + (−0.339 − 0.467i)3-s + (1.17 + 0.852i)4-s + (−0.0217 − 0.999i)5-s + (0.279 + 0.859i)6-s + (0.353 − 0.487i)7-s + (−0.414 − 0.570i)8-s + (−0.103 + 0.317i)9-s + (−0.451 + 1.49i)10-s + (−0.656 − 0.754i)11-s − 0.837i·12-s + (−0.235 − 0.0764i)13-s + (−0.762 + 0.554i)14-s + (−0.459 + 0.349i)15-s + (−0.106 − 0.329i)16-s + (−1.02 + 0.332i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.987 + 0.154i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -0.987 + 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0274924 - 0.352901i\)
\(L(\frac12)\) \(\approx\) \(0.0274924 - 0.352901i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 + (0.0486 + 2.23i)T \)
11 \( 1 + (2.17 + 2.50i)T \)
good2 \( 1 + (2.10 + 0.684i)T + (1.61 + 1.17i)T^{2} \)
7 \( 1 + (-0.936 + 1.28i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.847 + 0.275i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.22 - 1.37i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.72 - 3.43i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.27iT - 23T^{2} \)
29 \( 1 + (-4.28 - 3.11i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.75 + 8.47i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.25 + 4.47i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.38 + 2.46i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 6.39iT - 43T^{2} \)
47 \( 1 + (-4.98 - 6.85i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.16 + 1.35i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.82 + 4.23i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.627 - 1.93i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 13.6iT - 67T^{2} \)
71 \( 1 + (-0.356 - 1.09i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.91 - 5.38i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.51 + 4.65i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-12.5 + 4.09i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 8.06T + 89T^{2} \)
97 \( 1 + (-10.2 - 3.32i)T + (78.4 + 57.0i)T^{2} \)
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

−12.17384309287602766119707559178, −11.01404395493585647109878705167, −10.49546850304668226843082823928, −9.167042573569133323296367713791, −8.307504258485891838123898567624, −7.66975214863536038524491350867, −6.11230880598139835503168371398, −4.53235057121686093949659340745, −2.13343239404819403527089245163, −0.54050513819058992145594475300, 2.42026835235014904959290137684, 4.66679183778335721663987061613, 6.32482777855894251783443371457, 7.13254041332840057955692829008, 8.254606227250943717896526889318, 9.283887466688827572512282859948, 10.22833255527864787109737468467, 10.87064504891794972233873617196, 11.82174030934517921842608519929, 13.34797537298899116193949600604

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