Properties

Label 2-165-33.32-c5-0-18
Degree $2$
Conductor $165$
Sign $-0.859 - 0.510i$
Analytic cond. $26.4633$
Root an. cond. $5.14425$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.67·2-s + (4.39 + 14.9i)3-s + 12.5·4-s − 25i·5-s + (29.3 + 99.8i)6-s + 13.4i·7-s − 129.·8-s + (−204. + 131. i)9-s − 166. i·10-s + (273. + 293. i)11-s + (55.3 + 188. i)12-s + 375. i·13-s + 89.7i·14-s + (373. − 109. i)15-s − 1.26e3·16-s − 2.12e3·17-s + ⋯
L(s)  = 1  + 1.18·2-s + (0.282 + 0.959i)3-s + 0.393·4-s − 0.447i·5-s + (0.333 + 1.13i)6-s + 0.103i·7-s − 0.716·8-s + (−0.840 + 0.541i)9-s − 0.527i·10-s + (0.680 + 0.732i)11-s + (0.110 + 0.377i)12-s + 0.617i·13-s + 0.122i·14-s + (0.429 − 0.126i)15-s − 1.23·16-s − 1.78·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.510i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.859 - 0.510i$
Analytic conductor: \(26.4633\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :5/2),\ -0.859 - 0.510i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.222153195\)
\(L(\frac12)\) \(\approx\) \(2.222153195\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.39 - 14.9i)T \)
5 \( 1 + 25iT \)
11 \( 1 + (-273. - 293. i)T \)
good2 \( 1 - 6.67T + 32T^{2} \)
7 \( 1 - 13.4iT - 1.68e4T^{2} \)
13 \( 1 - 375. iT - 3.71e5T^{2} \)
17 \( 1 + 2.12e3T + 1.41e6T^{2} \)
19 \( 1 - 2.32e3iT - 2.47e6T^{2} \)
23 \( 1 - 164. iT - 6.43e6T^{2} \)
29 \( 1 + 655.T + 2.05e7T^{2} \)
31 \( 1 - 7.96e3T + 2.86e7T^{2} \)
37 \( 1 + 9.98e3T + 6.93e7T^{2} \)
41 \( 1 + 3.99e3T + 1.15e8T^{2} \)
43 \( 1 + 1.78e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.29e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.22e4iT - 4.18e8T^{2} \)
59 \( 1 - 5.19e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.62e4iT - 8.44e8T^{2} \)
67 \( 1 - 5.91e4T + 1.35e9T^{2} \)
71 \( 1 - 4.14e4iT - 1.80e9T^{2} \)
73 \( 1 - 423. iT - 2.07e9T^{2} \)
79 \( 1 - 4.54e4iT - 3.07e9T^{2} \)
83 \( 1 - 8.53e4T + 3.93e9T^{2} \)
89 \( 1 - 1.87e4iT - 5.58e9T^{2} \)
97 \( 1 - 6.13e4T + 8.58e9T^{2} \)
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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

−12.30494686690873400882538907777, −11.69819264667718314345216486570, −10.34969621429877740479205351858, −9.225903385038842748558410408310, −8.560858292072020180361045821769, −6.70811275854798064268296662917, −5.49150548460582152580678989165, −4.41365025703885029618843190215, −3.86493974656065514046828727623, −2.22445776742964688650464584512, 0.44208433142131919597939079933, 2.43157124681674848855937584698, 3.44096406645003955666745503466, 4.85435129652939807501783407569, 6.30148733850488675747757507242, 6.78383037836110989908086025388, 8.350665002991234418751802744981, 9.232558135318655113499986943281, 11.06206849003498953406800763750, 11.70197339891849097199332630492

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