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Properties

Label 2-165-5.4-c3-0-16
Degree $2$
Conductor $165$
Sign $0.998 + 0.0500i$
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  + 4.66i·2-s + 3i·3-s − 13.7·4-s + (0.559 − 11.1i)5-s − 13.9·6-s − 20.4i·7-s − 26.6i·8-s − 9·9-s + (52.0 + 2.60i)10-s − 11·11-s − 41.1i·12-s − 75.4i·13-s + 95.1·14-s + (33.4 + 1.67i)15-s + 14.4·16-s − 57.7i·17-s + ⋯
L(s)  = 1  + 1.64i·2-s + 0.577i·3-s − 1.71·4-s + (0.0500 − 0.998i)5-s − 0.951·6-s − 1.10i·7-s − 1.17i·8-s − 0.333·9-s + (1.64 + 0.0824i)10-s − 0.301·11-s − 0.990i·12-s − 1.61i·13-s + 1.81·14-s + (0.576 + 0.0288i)15-s + 0.225·16-s − 0.823i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0500i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.998 + 0.0500i$
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: \((2,\ 165,\ (\ :3/2),\ 0.998 + 0.0500i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.955679 - 0.0239278i\)
\(L(\frac12)\) \(\approx\) \(0.955679 - 0.0239278i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.559 + 11.1i)T \)
11 \( 1 + 11T \)
good2 \( 1 - 4.66iT - 8T^{2} \)
7 \( 1 + 20.4iT - 343T^{2} \)
13 \( 1 + 75.4iT - 2.19e3T^{2} \)
17 \( 1 + 57.7iT - 4.91e3T^{2} \)
19 \( 1 - 27.5T + 6.85e3T^{2} \)
23 \( 1 - 125. iT - 1.21e4T^{2} \)
29 \( 1 + 73.6T + 2.43e4T^{2} \)
31 \( 1 + 316.T + 2.97e4T^{2} \)
37 \( 1 + 71.5iT - 5.06e4T^{2} \)
41 \( 1 - 359.T + 6.89e4T^{2} \)
43 \( 1 - 380. iT - 7.95e4T^{2} \)
47 \( 1 + 576. iT - 1.03e5T^{2} \)
53 \( 1 + 252. iT - 1.48e5T^{2} \)
59 \( 1 - 552.T + 2.05e5T^{2} \)
61 \( 1 + 9.30T + 2.26e5T^{2} \)
67 \( 1 + 610. iT - 3.00e5T^{2} \)
71 \( 1 - 132.T + 3.57e5T^{2} \)
73 \( 1 + 220. iT - 3.89e5T^{2} \)
79 \( 1 + 9.99e2T + 4.93e5T^{2} \)
83 \( 1 - 229. iT - 5.71e5T^{2} \)
89 \( 1 - 178.T + 7.04e5T^{2} \)
97 \( 1 + 104. iT - 9.12e5T^{2} \)
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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

−12.91140665882445767645837654989, −11.22343678898727253777155275345, −9.949349366674528271079658238495, −9.082730468034745763244500846026, −7.914443848756968544749776110021, −7.33987984813996165062929779166, −5.64427956552451020306162100618, −5.09327968406918067356289038839, −3.79294416987310646773518337035, −0.44007397144152808874271054054, 1.86258163312498570321839552229, 2.65502933981127901800650998019, 4.05015709981403564534769298610, 5.84299102095687014782649299965, 7.10061894079057375720238601908, 8.693156700349260806129046447412, 9.511613997965487451004765436154, 10.73420519852483831031695170246, 11.38671727814002368357812934457, 12.22072793137178107812708268612

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