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Properties

Label 2-165-55.9-c1-0-5
Degree $2$
Conductor $165$
Sign $0.674 + 0.738i$
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  + (−0.283 − 0.0921i)2-s + (0.587 + 0.809i)3-s + (−1.54 − 1.12i)4-s + (0.882 − 2.05i)5-s + (−0.0921 − 0.283i)6-s + (1.36 − 1.87i)7-s + (0.685 + 0.943i)8-s + (−0.309 + 0.951i)9-s + (−0.439 + 0.501i)10-s + (−0.521 − 3.27i)11-s − 1.91i·12-s + (4.36 + 1.41i)13-s + (−0.559 + 0.406i)14-s + (2.18 − 0.493i)15-s + (1.07 + 3.30i)16-s + (−4.88 + 1.58i)17-s + ⋯
L(s)  = 1  + (−0.200 − 0.0651i)2-s + (0.339 + 0.467i)3-s + (−0.773 − 0.561i)4-s + (0.394 − 0.918i)5-s + (−0.0376 − 0.115i)6-s + (0.515 − 0.709i)7-s + (0.242 + 0.333i)8-s + (−0.103 + 0.317i)9-s + (−0.138 + 0.158i)10-s + (−0.157 − 0.987i)11-s − 0.551i·12-s + (1.21 + 0.393i)13-s + (−0.149 + 0.108i)14-s + (0.563 − 0.127i)15-s + (0.268 + 0.826i)16-s + (−1.18 + 0.385i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.674 + 0.738i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.674 + 0.738i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.987695 - 0.435516i\)
\(L(\frac12)\) \(\approx\) \(0.987695 - 0.435516i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.882 + 2.05i)T \)
11 \( 1 + (0.521 + 3.27i)T \)
good2 \( 1 + (0.283 + 0.0921i)T + (1.61 + 1.17i)T^{2} \)
7 \( 1 + (-1.36 + 1.87i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-4.36 - 1.41i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.88 - 1.58i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.60 + 3.34i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 6.02iT - 23T^{2} \)
29 \( 1 + (4.93 + 3.58i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.60 - 8.01i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.86 + 2.56i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.379 - 0.275i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 8.08iT - 43T^{2} \)
47 \( 1 + (-5.40 - 7.43i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.05 + 1.31i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.38 - 3.91i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.562 + 1.73i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 8.86iT - 67T^{2} \)
71 \( 1 + (0.756 + 2.32i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.33 + 7.34i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.99 + 12.2i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-7.93 + 2.57i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 3.73T + 89T^{2} \)
97 \( 1 + (5.31 + 1.72i)T + (78.4 + 57.0i)T^{2} \)
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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

−13.29968183819362229496121815036, −11.32138030257453517313436591747, −10.70957785361797391840024807818, −9.347346533183320643936888516208, −8.925791815588662544529588959624, −7.894237083448771184241313696683, −5.96209473095474789591386671245, −4.91152938510507813111563439799, −3.86217812788767079157244651825, −1.30219586344125999309510966018, 2.26748205836338133123939423404, 3.78833139032567719235733790216, 5.43374499166902769522710855460, 6.84228258577890936608869961348, 7.85492674507182242562866106139, 8.807586169323370257142387663052, 9.737030755132593250079165940110, 10.97987371881759952821712645639, 12.09823560512410067914292168762, 13.10259563044718871363721330173

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