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Properties

Label 2-165-165.164-c1-0-6
Degree $2$
Conductor $165$
Sign $0.205 - 0.978i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.50i·2-s + (1.53 − 0.796i)3-s − 0.267·4-s + (−1.12 + 1.93i)5-s + (1.19 + 2.31i)6-s − 2.12·7-s + 2.60i·8-s + (1.73 − 2.44i)9-s + (−2.90 − 1.69i)10-s + (3.27 + 0.517i)11-s + (−0.412 + 0.213i)12-s + 2.12·13-s − 3.20i·14-s + (−0.193 + 3.86i)15-s − 4.46·16-s − 4.11i·17-s + ⋯
L(s)  = 1  + 1.06i·2-s + (0.888 − 0.459i)3-s − 0.133·4-s + (−0.503 + 0.863i)5-s + (0.489 + 0.945i)6-s − 0.804·7-s + 0.922i·8-s + (0.577 − 0.816i)9-s + (−0.920 − 0.536i)10-s + (0.987 + 0.156i)11-s + (−0.118 + 0.0615i)12-s + 0.590·13-s − 0.857i·14-s + (−0.0500 + 0.998i)15-s − 1.11·16-s − 0.997i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.205 - 0.978i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12141 + 0.910565i\)
\(L(\frac12)\) \(\approx\) \(1.12141 + 0.910565i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.53 + 0.796i)T \)
5 \( 1 + (1.12 - 1.93i)T \)
11 \( 1 + (-3.27 - 0.517i)T \)
good2 \( 1 - 1.50iT - 2T^{2} \)
7 \( 1 + 2.12T + 7T^{2} \)
13 \( 1 - 2.12T + 13T^{2} \)
17 \( 1 + 4.11iT - 17T^{2} \)
19 \( 1 + 4.63iT - 19T^{2} \)
23 \( 1 + 5.32T + 23T^{2} \)
29 \( 1 + 8.95T + 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 - 1.16iT - 37T^{2} \)
41 \( 1 + 2.39T + 41T^{2} \)
43 \( 1 - 9.50T + 43T^{2} \)
47 \( 1 - 3.07T + 47T^{2} \)
53 \( 1 - 4.50T + 53T^{2} \)
59 \( 1 - 4.89iT - 59T^{2} \)
61 \( 1 - 3.39iT - 61T^{2} \)
67 \( 1 - 12.3iT - 67T^{2} \)
71 \( 1 + 13.3iT - 71T^{2} \)
73 \( 1 + 2.12T + 73T^{2} \)
79 \( 1 - 13.8iT - 79T^{2} \)
83 \( 1 - 9.33iT - 83T^{2} \)
89 \( 1 - 4.62iT - 89T^{2} \)
97 \( 1 + 1.16iT - 97T^{2} \)
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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

−13.49696253885707069045355376690, −12.08585630651298722595857411050, −11.19634674777275709961420858385, −9.661017208704317902064156993083, −8.715398419102545752745339458590, −7.51800971068083445301492061121, −6.91406236759319335886310752154, −6.09965746242592459771462074096, −3.94760367303602682200793232808, −2.60904230821185557397596556341, 1.72170706748314918862081434945, 3.60655931245249266237763523884, 4.00929884148087895920031569193, 6.12267839364989833451668483106, 7.72366120949241567034169009144, 8.864744324942900551719812365421, 9.625093754428314625794790277381, 10.54678046011614538670699844732, 11.68734810876722469765864131742, 12.59569607318666233320907382291

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