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Properties

Label 2-1320-165.164-c1-0-29
Degree $2$
Conductor $1320$
Sign $0.994 + 0.108i$
Analytic cond. $10.5402$
Root an. cond. $3.24657$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.72 − 0.203i)3-s + (−0.910 − 2.04i)5-s + 4.06·7-s + (2.91 + 0.698i)9-s + (3.04 + 1.32i)11-s + 0.487·13-s + (1.15 + 3.69i)15-s + 6.71i·17-s − 1.77i·19-s + (−6.99 − 0.826i)21-s − 4.87·23-s + (−3.34 + 3.71i)25-s + (−4.87 − 1.79i)27-s + 1.32·29-s + 6.21·31-s + ⋯
L(s)  = 1  + (−0.993 − 0.117i)3-s + (−0.407 − 0.913i)5-s + 1.53·7-s + (0.972 + 0.232i)9-s + (0.916 + 0.399i)11-s + 0.135·13-s + (0.297 + 0.954i)15-s + 1.62i·17-s − 0.407i·19-s + (−1.52 − 0.180i)21-s − 1.01·23-s + (−0.668 + 0.743i)25-s + (−0.938 − 0.345i)27-s + 0.246·29-s + 1.11·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.994 + 0.108i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ 0.994 + 0.108i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.413611628\)
\(L(\frac12)\) \(\approx\) \(1.413611628\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.72 + 0.203i)T \)
5 \( 1 + (0.910 + 2.04i)T \)
11 \( 1 + (-3.04 - 1.32i)T \)
good7 \( 1 - 4.06T + 7T^{2} \)
13 \( 1 - 0.487T + 13T^{2} \)
17 \( 1 - 6.71iT - 17T^{2} \)
19 \( 1 + 1.77iT - 19T^{2} \)
23 \( 1 + 4.87T + 23T^{2} \)
29 \( 1 - 1.32T + 29T^{2} \)
31 \( 1 - 6.21T + 31T^{2} \)
37 \( 1 - 7.96iT - 37T^{2} \)
41 \( 1 - 5.07T + 41T^{2} \)
43 \( 1 + 4.68T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 - 9.51T + 53T^{2} \)
59 \( 1 + 5.13iT - 59T^{2} \)
61 \( 1 - 2.61iT - 61T^{2} \)
67 \( 1 - 8.52iT - 67T^{2} \)
71 \( 1 + 6.00iT - 71T^{2} \)
73 \( 1 + 0.708T + 73T^{2} \)
79 \( 1 - 7.16iT - 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 - 15.9iT - 89T^{2} \)
97 \( 1 - 0.844iT - 97T^{2} \)
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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

−9.715708889161378157334311852763, −8.522485393986920307813532742602, −8.161520701430158673616824370367, −7.18280059583681559064971126458, −6.17213563460171617368207983039, −5.38258847575618144528830694879, −4.39602017804521172647150013927, −4.14208307530752680326684685462, −1.83471437326770704535395827507, −1.09181983622220327315078432935, 0.896986960011326710858802957728, 2.30556209492927747155340128829, 3.82284319635117170274857655428, 4.51729269799677347705339919144, 5.49784789796930173494891293475, 6.31032351259910888174695780761, 7.22378731888320070114634893517, 7.78770251779743788725601903763, 8.829366814201426945824238903123, 9.853131823413246218825698208765

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