Properties

Label 2-165-5.4-c3-0-2
Degree $2$
Conductor $165$
Sign $-0.307 + 0.951i$
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  + 3.70i·2-s + 3i·3-s − 5.72·4-s + (−10.6 − 3.44i)5-s − 11.1·6-s + 30.9i·7-s + 8.41i·8-s − 9·9-s + (12.7 − 39.4i)10-s + 11·11-s − 17.1i·12-s − 39.2i·13-s − 114.·14-s + (10.3 − 31.9i)15-s − 77.0·16-s − 83.0i·17-s + ⋯
L(s)  = 1  + 1.30i·2-s + 0.577i·3-s − 0.715·4-s + (−0.951 − 0.307i)5-s − 0.756·6-s + 1.66i·7-s + 0.372i·8-s − 0.333·9-s + (0.403 − 1.24i)10-s + 0.301·11-s − 0.413i·12-s − 0.838i·13-s − 2.18·14-s + (0.177 − 0.549i)15-s − 1.20·16-s − 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.307 + 0.951i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.307 + 0.951i$
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: \((2,\ 165,\ (\ :3/2),\ -0.307 + 0.951i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.479993 - 0.659911i\)
\(L(\frac12)\) \(\approx\) \(0.479993 - 0.659911i\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.6 + 3.44i)T \)
11 \( 1 - 11T \)
good2 \( 1 - 3.70iT - 8T^{2} \)
7 \( 1 - 30.9iT - 343T^{2} \)
13 \( 1 + 39.2iT - 2.19e3T^{2} \)
17 \( 1 + 83.0iT - 4.91e3T^{2} \)
19 \( 1 + 49.6T + 6.85e3T^{2} \)
23 \( 1 + 10.7iT - 1.21e4T^{2} \)
29 \( 1 + 198.T + 2.43e4T^{2} \)
31 \( 1 - 229.T + 2.97e4T^{2} \)
37 \( 1 - 348. iT - 5.06e4T^{2} \)
41 \( 1 + 459.T + 6.89e4T^{2} \)
43 \( 1 - 461. iT - 7.95e4T^{2} \)
47 \( 1 - 272. iT - 1.03e5T^{2} \)
53 \( 1 + 360. iT - 1.48e5T^{2} \)
59 \( 1 - 215.T + 2.05e5T^{2} \)
61 \( 1 - 367.T + 2.26e5T^{2} \)
67 \( 1 - 929. iT - 3.00e5T^{2} \)
71 \( 1 - 126.T + 3.57e5T^{2} \)
73 \( 1 + 509. iT - 3.89e5T^{2} \)
79 \( 1 + 719.T + 4.93e5T^{2} \)
83 \( 1 - 865. iT - 5.71e5T^{2} \)
89 \( 1 + 528.T + 7.04e5T^{2} \)
97 \( 1 - 858. iT - 9.12e5T^{2} \)
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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

−13.15538737486940151182256419902, −11.89765877222167951435765328489, −11.36677548945120782739129594578, −9.638642068035435598777815976034, −8.585149455824384785578503625006, −8.088068963429664137745411960394, −6.68575264889963282156745262439, −5.51878754945633895293949443837, −4.72763063615837407224047826818, −2.88837044120199990108340672630, 0.37461315167431367948848799921, 1.78949378783547865591115352008, 3.62453929417314269166450333720, 4.19172421071114909437703561773, 6.65824437332849308573100449468, 7.34958622244320823968934570904, 8.637211189939985474684420179777, 10.14549025700641210795424677869, 10.83902035833030653042993781438, 11.58231585922216809683022653379

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