Properties

Label 2-165-11.9-c1-0-6
Degree $2$
Conductor $165$
Sign $-0.981 + 0.192i$
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  + (0.758 − 2.33i)2-s + (−0.809 + 0.587i)3-s + (−3.26 − 2.36i)4-s + (−0.309 − 0.951i)5-s + (0.758 + 2.33i)6-s + (−2.65 − 1.93i)7-s + (−4.03 + 2.93i)8-s + (0.309 − 0.951i)9-s − 2.45·10-s + (2.96 − 1.47i)11-s + 4.03·12-s + (−0.0967 + 0.297i)13-s + (−6.53 + 4.74i)14-s + (0.809 + 0.587i)15-s + (1.29 + 3.98i)16-s + (1.54 + 4.75i)17-s + ⋯
L(s)  = 1  + (0.536 − 1.65i)2-s + (−0.467 + 0.339i)3-s + (−1.63 − 1.18i)4-s + (−0.138 − 0.425i)5-s + (0.309 + 0.953i)6-s + (−1.00 − 0.730i)7-s + (−1.42 + 1.03i)8-s + (0.103 − 0.317i)9-s − 0.776·10-s + (0.894 − 0.446i)11-s + 1.16·12-s + (−0.0268 + 0.0825i)13-s + (−1.74 + 1.26i)14-s + (0.208 + 0.151i)15-s + (0.323 + 0.996i)16-s + (0.374 + 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.981 + 0.192i$
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: \((2,\ 165,\ (\ :1/2),\ -0.981 + 0.192i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102390 - 1.05151i\)
\(L(\frac12)\) \(\approx\) \(0.102390 - 1.05151i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.96 + 1.47i)T \)
good2 \( 1 + (-0.758 + 2.33i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (2.65 + 1.93i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.0967 - 0.297i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.54 - 4.75i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-6.03 + 4.38i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.07T + 23T^{2} \)
29 \( 1 + (-4.07 - 2.96i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.06 + 3.28i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.13 + 1.54i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (8.77 - 6.37i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.51T + 43T^{2} \)
47 \( 1 + (-9.70 + 7.05i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.52 + 4.69i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.41 - 5.38i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.83 - 8.73i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 15.2T + 67T^{2} \)
71 \( 1 + (-0.949 - 2.92i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.00 - 5.08i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.67 + 5.14i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-5.02 - 15.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 1.62T + 89T^{2} \)
97 \( 1 + (-0.0692 + 0.213i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.12861012778694424971077985452, −11.55629346904288359917734427243, −10.46111032208628832702414573752, −9.804841894729485021608432733766, −8.832093102505381844901694938200, −6.80463306473484993856974795309, −5.36491894572619182328363524680, −4.09236183625956386239701851532, −3.27881105943847527635504645399, −0.996802772288967517521983178922, 3.38561193733536342553195425618, 5.02056400090550295996883660776, 6.03144301687022952365354579795, 6.82148417837969108039694699746, 7.62172752980393347425946717621, 8.976052115657130829684758337613, 9.981709508665628229338175745660, 11.87630336827339338470479782513, 12.36284770347997070138270078749, 13.66575861978809145214808044198

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