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Properties

Label 2-165-165.164-c1-0-2
Degree $2$
Conductor $165$
Sign $-0.917 - 0.397i$
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 + (−3.27 − 2.07i)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 + (−0.844 − 0.535i)15-s − 1.11·16-s − 0.997i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.917 - 0.397i$
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: \((2,\ 165,\ (\ :1/2),\ -0.917 - 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.183477 + 0.885349i\)
\(L(\frac12)\) \(\approx\) \(0.183477 + 0.885349i\)
\(L(\frac{3}{2})\) 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

−13.46293034859606547367540682530, −12.23253876226632563497634740242, −11.03663065953562295654449260462, −10.35889145457063180559837853203, −9.308679251402325350822745874039, −7.70093901197317874586161931693, −6.64455256023483084735829277897, −6.04585979869464195957725485934, −4.97443825244924781801631314868, −2.97893233812162270888722508714, 0.998470260566059661271444834021, 2.68907604924610659289550643512, 4.56094901043770993451836954972, 5.90153568903307848495402997264, 6.85894132010235689444875897800, 8.464646077704556845388819396108, 9.782071806597621940828433243615, 10.57478245076603318527660043085, 11.38641025133518425569059526769, 12.60288892416910281230056103224

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