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Properties

Label 2-165-11.9-c1-0-3
Degree $2$
Conductor $165$
Sign $0.931 - 0.362i$
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  + (−0.0646 + 0.198i)2-s + (0.809 − 0.587i)3-s + (1.58 + 1.14i)4-s + (0.309 + 0.951i)5-s + (0.0646 + 0.198i)6-s + (−0.395 − 0.287i)7-s + (−0.669 + 0.486i)8-s + (0.309 − 0.951i)9-s − 0.209·10-s + (−2.66 + 1.97i)11-s + 1.95·12-s + (1.00 − 3.10i)13-s + (0.0826 − 0.0600i)14-s + (0.809 + 0.587i)15-s + (1.15 + 3.55i)16-s + (−1.02 − 3.16i)17-s + ⋯
L(s)  = 1  + (−0.0456 + 0.140i)2-s + (0.467 − 0.339i)3-s + (0.791 + 0.574i)4-s + (0.138 + 0.425i)5-s + (0.0263 + 0.0811i)6-s + (−0.149 − 0.108i)7-s + (−0.236 + 0.171i)8-s + (0.103 − 0.317i)9-s − 0.0661·10-s + (−0.802 + 0.596i)11-s + 0.564·12-s + (0.280 − 0.861i)13-s + (0.0220 − 0.0160i)14-s + (0.208 + 0.151i)15-s + (0.288 + 0.889i)16-s + (−0.249 − 0.768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.931 - 0.362i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.931 - 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41168 + 0.265154i\)
\(L(\frac12)\) \(\approx\) \(1.41168 + 0.265154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (2.66 - 1.97i)T \)
good2 \( 1 + (0.0646 - 0.198i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (0.395 + 0.287i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.00 + 3.10i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.02 + 3.16i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.12 + 2.27i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.267T + 23T^{2} \)
29 \( 1 + (5.10 + 3.71i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.80 - 5.54i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.07 + 4.41i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.38 - 2.46i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.26T + 43T^{2} \)
47 \( 1 + (-4.51 + 3.28i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.353 - 1.08i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.65 - 2.65i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.88 - 11.9i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 8.97T + 67T^{2} \)
71 \( 1 + (-1.64 - 5.05i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.81 + 5.67i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.466 - 1.43i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-5.45 - 16.7i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + (5.04 - 15.5i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.94785870319235528186112860840, −11.96667977471441634109001122436, −10.93550308292936588271608412441, −9.927334393257833047907118088629, −8.574778300946015283854967184067, −7.48219121572564496171185204774, −6.93212576688395913752737880355, −5.45264507105222730758871756805, −3.43289735064648058583456450100, −2.35387126191931581029495034568, 1.90370741541760990583288768396, 3.49289657791096265642170623749, 5.21029763845718855289147445149, 6.27032221707727617928778677114, 7.64045632817515636748313097100, 8.821640662459545327476355982017, 9.801005194174767970348516647103, 10.73460073894408561659457357688, 11.61190543628903737405570341231, 12.79878599004185173957515768661

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