↑

Properties

Label 2-165-5.4-c3-0-6
Degree $2$
Conductor $165$
Sign $-0.996 - 0.0827i$
Analytic cond. $9.73531$
Root an. cond. $3.12014$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.57i·2-s − 3i·3-s − 4.80·4-s + (−0.925 + 11.1i)5-s + 10.7·6-s − 7.85i·7-s + 11.4i·8-s − 9·9-s + (−39.8 − 3.31i)10-s + 11·11-s + 14.4i·12-s + 74.4i·13-s + 28.1·14-s + (33.4 + 2.77i)15-s − 79.3·16-s − 45.2i·17-s + ⋯
L(s)  = 1  + 1.26i·2-s − 0.577i·3-s − 0.601·4-s + (−0.0827 + 0.996i)5-s + 0.730·6-s − 0.423i·7-s + 0.504i·8-s − 0.333·9-s + (−1.26 − 0.104i)10-s + 0.301·11-s + 0.346i·12-s + 1.58i·13-s + 0.536·14-s + (0.575 + 0.0477i)15-s − 1.23·16-s − 0.645i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0827i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.996 - 0.0827i$
Analytic conductor: \(9.73531\)
Root analytic conductor: \(3.12014\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :3/2),\ -0.996 - 0.0827i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0532048 + 1.28352i\)
\(L(\frac12)\) \(\approx\) \(0.0532048 + 1.28352i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 + (0.925 - 11.1i)T \)
11 \( 1 - 11T \)
good2 \( 1 - 3.57iT - 8T^{2} \)
7 \( 1 + 7.85iT - 343T^{2} \)
13 \( 1 - 74.4iT - 2.19e3T^{2} \)
17 \( 1 + 45.2iT - 4.91e3T^{2} \)
19 \( 1 + 126.T + 6.85e3T^{2} \)
23 \( 1 - 191. iT - 1.21e4T^{2} \)
29 \( 1 - 38.0T + 2.43e4T^{2} \)
31 \( 1 + 10.1T + 2.97e4T^{2} \)
37 \( 1 + 277. iT - 5.06e4T^{2} \)
41 \( 1 - 42.1T + 6.89e4T^{2} \)
43 \( 1 - 294. iT - 7.95e4T^{2} \)
47 \( 1 - 136. iT - 1.03e5T^{2} \)
53 \( 1 + 404. iT - 1.48e5T^{2} \)
59 \( 1 - 729.T + 2.05e5T^{2} \)
61 \( 1 - 388.T + 2.26e5T^{2} \)
67 \( 1 - 1.00e3iT - 3.00e5T^{2} \)
71 \( 1 - 300.T + 3.57e5T^{2} \)
73 \( 1 - 791. iT - 3.89e5T^{2} \)
79 \( 1 - 129.T + 4.93e5T^{2} \)
83 \( 1 - 757. iT - 5.71e5T^{2} \)
89 \( 1 - 545.T + 7.04e5T^{2} \)
97 \( 1 + 1.42e3iT - 9.12e5T^{2} \)
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

−13.23848235791983190385420391953, −11.65622669146469843764326082760, −11.11366112632150847792378860597, −9.564180755670791672169091164849, −8.379862353970798997690584365035, −7.14608743857437430997728591221, −6.87195008708312933834602159931, −5.78283926000829258559620546202, −4.10356174097508647666756169656, −2.17349193762564099418594669370, 0.57796842028555100762532037611, 2.32190152507944715154414876785, 3.76432055047686068225560837846, 4.84660641515096380888948206227, 6.24132439762798836498425126658, 8.269478892466817081893149599673, 8.957522819106399689300698220624, 10.23951223334310815366401197182, 10.68771782795091980939851195032, 12.01827920426677285822144183172

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