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Properties

Label 2-165-5.4-c3-0-3
Degree $2$
Conductor $165$
Sign $-0.981 + 0.193i$
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  + 4.00i·2-s − 3i·3-s − 8.05·4-s + (−2.16 − 10.9i)5-s + 12.0·6-s + 26.2i·7-s − 0.224i·8-s − 9·9-s + (43.9 − 8.68i)10-s − 11·11-s + 24.1i·12-s + 11.3i·13-s − 105.·14-s + (−32.9 + 6.50i)15-s − 63.5·16-s + 87.6i·17-s + ⋯
L(s)  = 1  + 1.41i·2-s − 0.577i·3-s − 1.00·4-s + (−0.193 − 0.981i)5-s + 0.817·6-s + 1.41i·7-s − 0.00990i·8-s − 0.333·9-s + (1.38 − 0.274i)10-s − 0.301·11-s + 0.581i·12-s + 0.241i·13-s − 2.00·14-s + (−0.566 + 0.111i)15-s − 0.992·16-s + 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.981 + 0.193i$
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: \((2,\ 165,\ (\ :3/2),\ -0.981 + 0.193i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0832118 - 0.850006i\)
\(L(\frac12)\) \(\approx\) \(0.0832118 - 0.850006i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.16 + 10.9i)T \)
11 \( 1 + 11T \)
good2 \( 1 - 4.00iT - 8T^{2} \)
7 \( 1 - 26.2iT - 343T^{2} \)
13 \( 1 - 11.3iT - 2.19e3T^{2} \)
17 \( 1 - 87.6iT - 4.91e3T^{2} \)
19 \( 1 + 148.T + 6.85e3T^{2} \)
23 \( 1 - 12.7iT - 1.21e4T^{2} \)
29 \( 1 - 37.2T + 2.43e4T^{2} \)
31 \( 1 + 30.4T + 2.97e4T^{2} \)
37 \( 1 - 122. iT - 5.06e4T^{2} \)
41 \( 1 - 444.T + 6.89e4T^{2} \)
43 \( 1 + 36.2iT - 7.95e4T^{2} \)
47 \( 1 - 78.5iT - 1.03e5T^{2} \)
53 \( 1 - 342. iT - 1.48e5T^{2} \)
59 \( 1 + 377.T + 2.05e5T^{2} \)
61 \( 1 + 690.T + 2.26e5T^{2} \)
67 \( 1 + 696. iT - 3.00e5T^{2} \)
71 \( 1 - 1.01e3T + 3.57e5T^{2} \)
73 \( 1 - 889. iT - 3.89e5T^{2} \)
79 \( 1 - 858.T + 4.93e5T^{2} \)
83 \( 1 + 115. iT - 5.71e5T^{2} \)
89 \( 1 + 553.T + 7.04e5T^{2} \)
97 \( 1 - 336. iT - 9.12e5T^{2} \)
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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

−12.79783595988379905140850390717, −12.36692197201060339830680315546, −11.05759267372977119248040150392, −9.166326979797481014666077934853, −8.503490589310540010348180228623, −7.84736512779706578997736277572, −6.34143129968856972581701344256, −5.74362478361409988830106328877, −4.53724026531256945890582102729, −2.09698344676926362900406906832, 0.37676581177375923113794615757, 2.46893575930620065398594452030, 3.63366146568387712378115191210, 4.51612869618840277617695651130, 6.56721014873617416117312612098, 7.67367940437240987852230591213, 9.299807033957479965070428704894, 10.36346303843456431802744815979, 10.70437220224047617816343693592, 11.43584657473988216553429714751

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