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Properties

Label 2-165-11.5-c1-0-3
Degree $2$
Conductor $165$
Sign $0.837 - 0.546i$
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  + (0.359 + 1.10i)2-s + (−0.809 − 0.587i)3-s + (0.525 − 0.381i)4-s + (−0.309 + 0.951i)5-s + (0.359 − 1.10i)6-s + (3.46 − 2.51i)7-s + (2.49 + 1.80i)8-s + (0.309 + 0.951i)9-s − 1.16·10-s + (−3.15 + 1.00i)11-s − 0.649·12-s + (1.59 + 4.91i)13-s + (4.03 + 2.92i)14-s + (0.809 − 0.587i)15-s + (−0.704 + 2.16i)16-s + (1.54 − 4.75i)17-s + ⋯
L(s)  = 1  + (0.253 + 0.781i)2-s + (−0.467 − 0.339i)3-s + (0.262 − 0.190i)4-s + (−0.138 + 0.425i)5-s + (0.146 − 0.451i)6-s + (1.31 − 0.952i)7-s + (0.880 + 0.639i)8-s + (0.103 + 0.317i)9-s − 0.367·10-s + (−0.952 + 0.304i)11-s − 0.187·12-s + (0.442 + 1.36i)13-s + (1.07 + 0.782i)14-s + (0.208 − 0.151i)15-s + (−0.176 + 0.541i)16-s + (0.374 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.837 - 0.546i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.837 - 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28775 + 0.382758i\)
\(L(\frac12)\) \(\approx\) \(1.28775 + 0.382758i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.15 - 1.00i)T \)
good2 \( 1 + (-0.359 - 1.10i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (-3.46 + 2.51i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.59 - 4.91i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.54 + 4.75i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.53 + 3.29i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 0.219T + 23T^{2} \)
29 \( 1 + (5.19 - 3.77i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.874 + 2.69i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.17 - 2.30i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.74 + 3.44i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.90T + 43T^{2} \)
47 \( 1 + (-0.192 - 0.139i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.783 - 2.41i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.36 + 4.62i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.50 - 4.62i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + (3.08 - 9.49i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (11.7 - 8.55i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.47 - 7.61i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.87 + 11.9i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 2.56T + 89T^{2} \)
97 \( 1 + (-0.621 - 1.91i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.27060537971907756192194973784, −11.55615378649461800795366882146, −11.15015096358005468169903584083, −10.22913585250767233746091511049, −8.381900637455137516193802052820, −7.26073460959360464878457763276, −6.88361768602335330114666689524, −5.34349128393525413109899329100, −4.45928757063915650399965530766, −1.90735037684957893618664315576, 1.88268777715664349553575339617, 3.56488978882370122432826110871, 4.99014522684934311030507212660, 5.90960818620160315228753971683, 7.920336182909557073702448529308, 8.422315226537699732523262951968, 10.26953816262382043661028155339, 10.83437731229539458560514589487, 11.74063748445128960238649709720, 12.55384709585567377664425835215

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