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Properties

Label 2-165-15.2-c1-0-13
Degree $2$
Conductor $165$
Sign $0.781 + 0.623i$
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  + (−1.31 + 1.31i)2-s + (0.825 − 1.52i)3-s − 1.47i·4-s + (1.49 − 1.66i)5-s + (0.918 + 3.09i)6-s + (−2.09 − 2.09i)7-s + (−0.695 − 0.695i)8-s + (−1.63 − 2.51i)9-s + (0.228 + 4.16i)10-s − i·11-s + (−2.24 − 1.21i)12-s + (0.161 − 0.161i)13-s + 5.51·14-s + (−1.30 − 3.64i)15-s + 4.77·16-s + (3.37 − 3.37i)17-s + ⋯
L(s)  = 1  + (−0.931 + 0.931i)2-s + (0.476 − 0.879i)3-s − 0.735i·4-s + (0.667 − 0.744i)5-s + (0.375 + 1.26i)6-s + (−0.791 − 0.791i)7-s + (−0.245 − 0.245i)8-s + (−0.545 − 0.837i)9-s + (0.0721 + 1.31i)10-s − 0.301i·11-s + (−0.647 − 0.350i)12-s + (0.0448 − 0.0448i)13-s + 1.47·14-s + (−0.336 − 0.941i)15-s + 1.19·16-s + (0.818 − 0.818i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.781 + 0.623i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.781 + 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.760371 - 0.266299i\)
\(L(\frac12)\) \(\approx\) \(0.760371 - 0.266299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.825 + 1.52i)T \)
5 \( 1 + (-1.49 + 1.66i)T \)
11 \( 1 + iT \)
good2 \( 1 + (1.31 - 1.31i)T - 2iT^{2} \)
7 \( 1 + (2.09 + 2.09i)T + 7iT^{2} \)
13 \( 1 + (-0.161 + 0.161i)T - 13iT^{2} \)
17 \( 1 + (-3.37 + 3.37i)T - 17iT^{2} \)
19 \( 1 - 8.52iT - 19T^{2} \)
23 \( 1 + (-3.15 - 3.15i)T + 23iT^{2} \)
29 \( 1 - 3.01T + 29T^{2} \)
31 \( 1 - 5.01T + 31T^{2} \)
37 \( 1 + (4.74 + 4.74i)T + 37iT^{2} \)
41 \( 1 - 4.49iT - 41T^{2} \)
43 \( 1 + (-0.336 + 0.336i)T - 43iT^{2} \)
47 \( 1 + (6.15 - 6.15i)T - 47iT^{2} \)
53 \( 1 + (-1.62 - 1.62i)T + 53iT^{2} \)
59 \( 1 - 5.16T + 59T^{2} \)
61 \( 1 + 3.39T + 61T^{2} \)
67 \( 1 + (-9.74 - 9.74i)T + 67iT^{2} \)
71 \( 1 + 2.23iT - 71T^{2} \)
73 \( 1 + (-5.43 + 5.43i)T - 73iT^{2} \)
79 \( 1 + 5.93iT - 79T^{2} \)
83 \( 1 + (-0.365 - 0.365i)T + 83iT^{2} \)
89 \( 1 + 3.70T + 89T^{2} \)
97 \( 1 + (-4.94 - 4.94i)T + 97iT^{2} \)
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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

−12.83908309197287441378760639730, −12.06084178636058554799907098733, −10.09005560056152996622083188530, −9.481158076828578887691784159292, −8.416832912684829029616958810691, −7.63637543217137272603451931729, −6.61272207518042549989597840539, −5.71072680052253979848840245612, −3.39140504760558368890012181528, −1.04929538480923639996588369965, 2.40116006432042086284843625560, 3.19562832074609626161821128392, 5.25250520121301244122648506359, 6.61616694463148681166951866240, 8.413737282496591773214903731533, 9.218112779814282152290480354036, 9.940631847809710706168087603292, 10.58967867291354541125698861302, 11.53333535471262744345647499241, 12.75789004792968118533084668960

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