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Properties

Label 2-165-5.4-c3-0-7
Degree $2$
Conductor $165$
Sign $-0.908 - 0.418i$
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  + 1.35i·2-s + 3i·3-s + 6.17·4-s + (−4.68 + 10.1i)5-s − 4.05·6-s + 5.74i·7-s + 19.1i·8-s − 9·9-s + (−13.7 − 6.32i)10-s − 11·11-s + 18.5i·12-s − 18.0i·13-s − 7.77·14-s + (−30.4 − 14.0i)15-s + 23.4·16-s + 58.2i·17-s + ⋯
L(s)  = 1  + 0.478i·2-s + 0.577i·3-s + 0.771·4-s + (−0.418 + 0.908i)5-s − 0.276·6-s + 0.310i·7-s + 0.847i·8-s − 0.333·9-s + (−0.434 − 0.200i)10-s − 0.301·11-s + 0.445i·12-s − 0.384i·13-s − 0.148·14-s + (−0.524 − 0.241i)15-s + 0.366·16-s + 0.830i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.908 - 0.418i$
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: \((2,\ 165,\ (\ :3/2),\ -0.908 - 0.418i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.343944 + 1.56784i\)
\(L(\frac12)\) \(\approx\) \(0.343944 + 1.56784i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.68 - 10.1i)T \)
11 \( 1 + 11T \)
good2 \( 1 - 1.35iT - 8T^{2} \)
7 \( 1 - 5.74iT - 343T^{2} \)
13 \( 1 + 18.0iT - 2.19e3T^{2} \)
17 \( 1 - 58.2iT - 4.91e3T^{2} \)
19 \( 1 + 14.3T + 6.85e3T^{2} \)
23 \( 1 - 27.6iT - 1.21e4T^{2} \)
29 \( 1 + 88.4T + 2.43e4T^{2} \)
31 \( 1 + 48.3T + 2.97e4T^{2} \)
37 \( 1 + 8.79iT - 5.06e4T^{2} \)
41 \( 1 + 53.2T + 6.89e4T^{2} \)
43 \( 1 - 232. iT - 7.95e4T^{2} \)
47 \( 1 - 65.2iT - 1.03e5T^{2} \)
53 \( 1 - 205. iT - 1.48e5T^{2} \)
59 \( 1 - 781.T + 2.05e5T^{2} \)
61 \( 1 - 744.T + 2.26e5T^{2} \)
67 \( 1 + 603. iT - 3.00e5T^{2} \)
71 \( 1 - 771.T + 3.57e5T^{2} \)
73 \( 1 - 377. iT - 3.89e5T^{2} \)
79 \( 1 - 202.T + 4.93e5T^{2} \)
83 \( 1 - 964. iT - 5.71e5T^{2} \)
89 \( 1 - 387.T + 7.04e5T^{2} \)
97 \( 1 + 1.22e3iT - 9.12e5T^{2} \)
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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

−12.67711624440001415423658598841, −11.51921120244875055657624832099, −10.86309172758640297422349777438, −9.992212071009087903502660935760, −8.452868103372003121083966109962, −7.54319258175109690357933897289, −6.43787496776457752068448854292, −5.43608289720864756616274987010, −3.68814666217135721338182250959, −2.41717118954405999329467564911, 0.72206694793231225683922860088, 2.18034405990190887935964657656, 3.79613742348018230851624907739, 5.34719751146677654434107663885, 6.80456886037589816026997795265, 7.64656666299009877517713759979, 8.820032699504718317651306713331, 10.04333715441691984095478698063, 11.25181308187136655690049336836, 11.90013159627469458362413372133

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