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Properties

Label 2-165-5.4-c3-0-14
Degree $2$
Conductor $165$
Sign $0.0910 + 0.995i$
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  − 1.41i·2-s + 3i·3-s + 6.00·4-s + (−11.1 + 1.01i)5-s + 4.23·6-s − 15.2i·7-s − 19.7i·8-s − 9·9-s + (1.43 + 15.7i)10-s + 11·11-s + 18.0i·12-s − 58.7i·13-s − 21.5·14-s + (−3.05 − 33.4i)15-s + 20.1·16-s − 55.1i·17-s + ⋯
L(s)  = 1  − 0.499i·2-s + 0.577i·3-s + 0.750·4-s + (−0.995 + 0.0910i)5-s + 0.288·6-s − 0.825i·7-s − 0.873i·8-s − 0.333·9-s + (0.0454 + 0.497i)10-s + 0.301·11-s + 0.433i·12-s − 1.25i·13-s − 0.411·14-s + (−0.0525 − 0.574i)15-s + 0.314·16-s − 0.786i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0910 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0910 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.0910 + 0.995i$
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: \((2,\ 165,\ (\ :3/2),\ 0.0910 + 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.17598 - 1.07339i\)
\(L(\frac12)\) \(\approx\) \(1.17598 - 1.07339i\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.1 - 1.01i)T \)
11 \( 1 - 11T \)
good2 \( 1 + 1.41iT - 8T^{2} \)
7 \( 1 + 15.2iT - 343T^{2} \)
13 \( 1 + 58.7iT - 2.19e3T^{2} \)
17 \( 1 + 55.1iT - 4.91e3T^{2} \)
19 \( 1 - 146.T + 6.85e3T^{2} \)
23 \( 1 + 4.30iT - 1.21e4T^{2} \)
29 \( 1 + 34.2T + 2.43e4T^{2} \)
31 \( 1 + 255.T + 2.97e4T^{2} \)
37 \( 1 + 323. iT - 5.06e4T^{2} \)
41 \( 1 + 186.T + 6.89e4T^{2} \)
43 \( 1 + 119. iT - 7.95e4T^{2} \)
47 \( 1 - 484. iT - 1.03e5T^{2} \)
53 \( 1 + 44.8iT - 1.48e5T^{2} \)
59 \( 1 + 263.T + 2.05e5T^{2} \)
61 \( 1 - 27.5T + 2.26e5T^{2} \)
67 \( 1 - 1.05e3iT - 3.00e5T^{2} \)
71 \( 1 - 81.3T + 3.57e5T^{2} \)
73 \( 1 - 623. iT - 3.89e5T^{2} \)
79 \( 1 - 896.T + 4.93e5T^{2} \)
83 \( 1 + 575. iT - 5.71e5T^{2} \)
89 \( 1 - 1.06e3T + 7.04e5T^{2} \)
97 \( 1 - 544. iT - 9.12e5T^{2} \)
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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

−11.86933107484626760009097343715, −11.10269762864651576724698011830, −10.41425285501888020269429346906, −9.340641092601198878793018458074, −7.71324751933709060749240391224, −7.15528776228236484927033600594, −5.43409377455982135242167332747, −3.87082801885771273717951371578, −3.05469643962427221768807165438, −0.76482628713316200245859290942, 1.74395940290113141591669064014, 3.38282064906684338421964062002, 5.20693247047529544824955961098, 6.44663077748568719371885704366, 7.30980739350096710968788067864, 8.217975980562313291877971132178, 9.242723711697072827735700862876, 10.98913698492741616306920230926, 11.87301623404822683396697178611, 12.15614497709324829914034075495

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