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Properties

Label 2-165-55.49-c1-0-4
Degree $2$
Conductor $165$
Sign $0.648 - 0.760i$
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  + (0.464 − 0.150i)2-s + (−0.587 + 0.809i)3-s + (−1.42 + 1.03i)4-s + (1.35 − 1.77i)5-s + (−0.150 + 0.464i)6-s + (3.00 + 4.13i)7-s + (−1.07 + 1.48i)8-s + (−0.309 − 0.951i)9-s + (0.362 − 1.02i)10-s + (2.66 + 1.97i)11-s − 1.76i·12-s + (−1.72 + 0.561i)13-s + (2.01 + 1.46i)14-s + (0.639 + 2.14i)15-s + (0.811 − 2.49i)16-s + (−0.608 − 0.197i)17-s + ⋯
L(s)  = 1  + (0.328 − 0.106i)2-s + (−0.339 + 0.467i)3-s + (−0.712 + 0.517i)4-s + (0.607 − 0.794i)5-s + (−0.0616 + 0.189i)6-s + (1.13 + 1.56i)7-s + (−0.381 + 0.525i)8-s + (−0.103 − 0.317i)9-s + (0.114 − 0.325i)10-s + (0.804 + 0.594i)11-s − 0.508i·12-s + (−0.479 + 0.155i)13-s + (0.539 + 0.391i)14-s + (0.165 + 0.553i)15-s + (0.202 − 0.624i)16-s + (−0.147 − 0.0479i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.648 - 0.760i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.648 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10379 + 0.509285i\)
\(L(\frac12)\) \(\approx\) \(1.10379 + 0.509285i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (-1.35 + 1.77i)T \)
11 \( 1 + (-2.66 - 1.97i)T \)
good2 \( 1 + (-0.464 + 0.150i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + (-3.00 - 4.13i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.72 - 0.561i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.608 + 0.197i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.55 + 2.58i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.37iT - 23T^{2} \)
29 \( 1 + (-6.11 + 4.44i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.681 - 2.09i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.51 + 4.83i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.80 - 1.31i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.38iT - 43T^{2} \)
47 \( 1 + (0.890 - 1.22i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (8.72 - 2.83i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.34 + 3.15i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.24 + 6.91i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 7.25iT - 67T^{2} \)
71 \( 1 + (-1.70 + 5.23i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.19 - 5.77i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.588 + 1.81i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.153 - 0.0500i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 1.19T + 89T^{2} \)
97 \( 1 + (13.5 - 4.41i)T + (78.4 - 57.0i)T^{2} \)
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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

−12.53214715222048450656653893696, −12.29590096642336400709382559385, −11.27230390666222100318198614407, −9.653585421762338673948298866536, −8.912770241260260598670970358018, −8.253833357121200309534917349407, −6.19632779987212399104539941989, −4.94142779507932225115845508213, −4.54267496244929716460097354733, −2.30803147759910910136686770523, 1.37004884754355292871337523848, 3.82501645131182438188899942065, 5.05520252089036781235848243905, 6.28974307944349089196956374296, 7.21326391587925819656136316876, 8.488354062993997487572040156208, 9.991993060723757823298984706988, 10.64237868834700455835725318726, 11.57715184108124324438410964715, 13.01370936030324555154126348781

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