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Properties

Label 2-165-5.4-c3-0-22
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

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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 \( 1 - 1.42e3iT - 9.12e5T^{2} \)
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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

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

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