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Properties

Label 2-165-5.4-c5-0-18
Degree $2$
Conductor $165$
Sign $0.923 - 0.384i$
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  − 3.33i·2-s − 9i·3-s + 20.8·4-s + (21.4 + 51.6i)5-s − 30.0·6-s + 103. i·7-s − 176. i·8-s − 81·9-s + (172. − 71.6i)10-s + 121·11-s − 187. i·12-s + 357. i·13-s + 345.·14-s + (464. − 193. i)15-s + 78.7·16-s + 2.06e3i·17-s + ⋯
L(s)  = 1  − 0.589i·2-s − 0.577i·3-s + 0.651·4-s + (0.384 + 0.923i)5-s − 0.340·6-s + 0.798i·7-s − 0.974i·8-s − 0.333·9-s + (0.544 − 0.226i)10-s + 0.301·11-s − 0.376i·12-s + 0.587i·13-s + 0.470·14-s + (0.533 − 0.221i)15-s + 0.0769·16-s + 1.73i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.923 - 0.384i$
Analytic conductor: \(26.4633\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :5/2),\ 0.923 - 0.384i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.315379760\)
\(L(\frac12)\) \(\approx\) \(2.315379760\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9iT \)
5 \( 1 + (-21.4 - 51.6i)T \)
11 \( 1 - 121T \)
good2 \( 1 + 3.33iT - 32T^{2} \)
7 \( 1 - 103. iT - 1.68e4T^{2} \)
13 \( 1 - 357. iT - 3.71e5T^{2} \)
17 \( 1 - 2.06e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.32e3T + 2.47e6T^{2} \)
23 \( 1 - 2.49e3iT - 6.43e6T^{2} \)
29 \( 1 + 1.65e3T + 2.05e7T^{2} \)
31 \( 1 - 4.69e3T + 2.86e7T^{2} \)
37 \( 1 - 5.21e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.35e4T + 1.15e8T^{2} \)
43 \( 1 + 1.07e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.92e3iT - 2.29e8T^{2} \)
53 \( 1 - 2.15e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.54e3T + 7.14e8T^{2} \)
61 \( 1 - 3.64e3T + 8.44e8T^{2} \)
67 \( 1 + 5.54e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.49e3T + 1.80e9T^{2} \)
73 \( 1 + 3.35e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.91e3T + 3.07e9T^{2} \)
83 \( 1 - 2.05e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.11e5T + 5.58e9T^{2} \)
97 \( 1 + 5.82e4iT - 8.58e9T^{2} \)
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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

−11.93342390000239908888380291106, −11.07991084250730365085752303655, −10.26214552521989560875973556931, −9.054081819655425650029885124984, −7.69482163186193926732744359545, −6.51244297208691206389161220721, −5.96991714406745881139817093832, −3.72292495147048369873077979723, −2.42452483815603115578804186732, −1.62007734960693401423799333604, 0.73352608652588826333737046270, 2.55198493882752651906235517965, 4.33248587415671373505872867117, 5.35990343172725037330733584960, 6.51127964012430454975605993077, 7.67120078304064702506358125636, 8.715164447771539302592830549677, 9.817596623745795092837537530748, 10.80264536429737473131682953607, 11.78315080921837848326130291208

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