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Properties

Label 2-165-5.4-c3-0-15
Degree $2$
Conductor $165$
Sign $0.910 + 0.413i$
Analytic cond. $9.73531$
Root an. cond. $3.12014$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.20i·2-s − 3i·3-s + 3.12·4-s + (4.62 − 10.1i)5-s + 6.62·6-s − 1.50i·7-s + 24.5i·8-s − 9·9-s + (22.4 + 10.1i)10-s + 11·11-s − 9.38i·12-s − 68.3i·13-s + 3.33·14-s + (−30.5 − 13.8i)15-s − 29.1·16-s − 113. i·17-s + ⋯
L(s)  = 1  + 0.780i·2-s − 0.577i·3-s + 0.391·4-s + (0.413 − 0.910i)5-s + 0.450·6-s − 0.0815i·7-s + 1.08i·8-s − 0.333·9-s + (0.710 + 0.322i)10-s + 0.301·11-s − 0.225i·12-s − 1.45i·13-s + 0.0636·14-s + (−0.525 − 0.238i)15-s − 0.455·16-s − 1.61i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.910 + 0.413i$
Analytic conductor: \(9.73531\)
Root analytic conductor: \(3.12014\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :3/2),\ 0.910 + 0.413i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.03479 - 0.440176i\)
\(L(\frac12)\) \(\approx\) \(2.03479 - 0.440176i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 + (-4.62 + 10.1i)T \)
11 \( 1 - 11T \)
good2 \( 1 - 2.20iT - 8T^{2} \)
7 \( 1 + 1.50iT - 343T^{2} \)
13 \( 1 + 68.3iT - 2.19e3T^{2} \)
17 \( 1 + 113. iT - 4.91e3T^{2} \)
19 \( 1 - 72.3T + 6.85e3T^{2} \)
23 \( 1 - 144. iT - 1.21e4T^{2} \)
29 \( 1 - 133.T + 2.43e4T^{2} \)
31 \( 1 - 177.T + 2.97e4T^{2} \)
37 \( 1 + 39.8iT - 5.06e4T^{2} \)
41 \( 1 + 366.T + 6.89e4T^{2} \)
43 \( 1 + 427. iT - 7.95e4T^{2} \)
47 \( 1 - 340. iT - 1.03e5T^{2} \)
53 \( 1 - 659. iT - 1.48e5T^{2} \)
59 \( 1 - 525.T + 2.05e5T^{2} \)
61 \( 1 + 462.T + 2.26e5T^{2} \)
67 \( 1 + 514. iT - 3.00e5T^{2} \)
71 \( 1 + 848.T + 3.57e5T^{2} \)
73 \( 1 - 987. iT - 3.89e5T^{2} \)
79 \( 1 + 442.T + 4.93e5T^{2} \)
83 \( 1 - 603. iT - 5.71e5T^{2} \)
89 \( 1 + 1.00e3T + 7.04e5T^{2} \)
97 \( 1 - 468. iT - 9.12e5T^{2} \)
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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

−12.23843027114884765949706030978, −11.62225568099577659553557205476, −10.17362315720302194960046242877, −8.960694023445178440389365578540, −7.902555543393291016288355753230, −7.12872675375537911237367185759, −5.80967064068176399397192259189, −5.10451488020851045490322188864, −2.81071550691361458844015706423, −1.04979174364760171567819591926, 1.80051404134834496574961651442, 3.09291540688947688795473092717, 4.28181945623050946490488221794, 6.18602166013850815381746146813, 6.88503078322023100413430477250, 8.584897199296443472119956451878, 9.902782394345357212853682360171, 10.36984805938847432346913358410, 11.41185826424545761089809571712, 12.03047571702099170412890978092

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