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Properties

Label 2-165-5.4-c3-0-20
Degree $2$
Conductor $165$
Sign $0.277 + 0.960i$
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  − 0.352i·2-s − 3i·3-s + 7.87·4-s + (10.7 − 3.10i)5-s − 1.05·6-s − 19.8i·7-s − 5.59i·8-s − 9·9-s + (−1.09 − 3.78i)10-s + 11·11-s − 23.6i·12-s + 25.5i·13-s − 6.99·14-s + (−9.31 − 32.2i)15-s + 61.0·16-s + 77.3i·17-s + ⋯
L(s)  = 1  − 0.124i·2-s − 0.577i·3-s + 0.984·4-s + (0.960 − 0.277i)5-s − 0.0719·6-s − 1.07i·7-s − 0.247i·8-s − 0.333·9-s + (−0.0346 − 0.119i)10-s + 0.301·11-s − 0.568i·12-s + 0.544i·13-s − 0.133·14-s + (−0.160 − 0.554i)15-s + 0.953·16-s + 1.10i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.277 + 0.960i$
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: \((2,\ 165,\ (\ :3/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.92734 - 1.44894i\)
\(L(\frac12)\) \(\approx\) \(1.92734 - 1.44894i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.7 + 3.10i)T \)
11 \( 1 - 11T \)
good2 \( 1 + 0.352iT - 8T^{2} \)
7 \( 1 + 19.8iT - 343T^{2} \)
13 \( 1 - 25.5iT - 2.19e3T^{2} \)
17 \( 1 - 77.3iT - 4.91e3T^{2} \)
19 \( 1 + 96.1T + 6.85e3T^{2} \)
23 \( 1 + 173. iT - 1.21e4T^{2} \)
29 \( 1 - 189.T + 2.43e4T^{2} \)
31 \( 1 + 275.T + 2.97e4T^{2} \)
37 \( 1 + 269. iT - 5.06e4T^{2} \)
41 \( 1 - 306.T + 6.89e4T^{2} \)
43 \( 1 - 477. iT - 7.95e4T^{2} \)
47 \( 1 - 257. iT - 1.03e5T^{2} \)
53 \( 1 - 495. iT - 1.48e5T^{2} \)
59 \( 1 + 102.T + 2.05e5T^{2} \)
61 \( 1 + 585.T + 2.26e5T^{2} \)
67 \( 1 + 474. iT - 3.00e5T^{2} \)
71 \( 1 - 453.T + 3.57e5T^{2} \)
73 \( 1 - 655. iT - 3.89e5T^{2} \)
79 \( 1 + 482.T + 4.93e5T^{2} \)
83 \( 1 - 523. iT - 5.71e5T^{2} \)
89 \( 1 - 750.T + 7.04e5T^{2} \)
97 \( 1 - 572. iT - 9.12e5T^{2} \)
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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

−12.46895784047571017888444412285, −10.88149098266790050646835354020, −10.53404571816439166023702116577, −9.110788718652964293003723046735, −7.84333013136191731380881240337, −6.64407836381918711988305257438, −6.12355410611255905839643032033, −4.23851852517338934134318109786, −2.39136127690927239848987522382, −1.23009925026891333598775666200, 2.03121693342001791795399043490, 3.13612091066459653405592651637, 5.26220496222884729149507600554, 6.00926730155861420511083022653, 7.14459841372930831617569674178, 8.639021781611077911417032760772, 9.605181654552690471792690419554, 10.56557423638720895854811625222, 11.48605317476597456245891168913, 12.39882237805231632860104955745

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