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Properties

Label 2-1155-33.32-c1-0-63
Degree $2$
Conductor $1155$
Sign $0.492 - 0.870i$
Analytic cond. $9.22272$
Root an. cond. $3.03689$
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  + 2.41·2-s + (1 + 1.41i)3-s + 3.82·4-s + i·5-s + (2.41 + 3.41i)6-s − i·7-s + 4.41·8-s + (−1.00 + 2.82i)9-s + 2.41i·10-s + (1.41 + 3i)11-s + (3.82 + 5.41i)12-s − 2.41i·14-s + (−1.41 + i)15-s + 2.99·16-s − 1.17·17-s + (−2.41 + 6.82i)18-s + ⋯
L(s)  = 1  + 1.70·2-s + (0.577 + 0.816i)3-s + 1.91·4-s + 0.447i·5-s + (0.985 + 1.39i)6-s − 0.377i·7-s + 1.56·8-s + (−0.333 + 0.942i)9-s + 0.763i·10-s + (0.426 + 0.904i)11-s + (1.10 + 1.56i)12-s − 0.645i·14-s + (−0.365 + 0.258i)15-s + 0.749·16-s − 0.284·17-s + (−0.569 + 1.60i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.492 - 0.870i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.492 - 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(5.189742509\)
\(L(\frac12)\) \(\approx\) \(5.189742509\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1 - 1.41i)T \)
5 \( 1 - iT \)
7 \( 1 + iT \)
11 \( 1 + (-1.41 - 3i)T \)
good2 \( 1 - 2.41T + 2T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 1.17T + 17T^{2} \)
19 \( 1 + 4.82iT - 19T^{2} \)
23 \( 1 + 4.82iT - 23T^{2} \)
29 \( 1 - 5.65T + 29T^{2} \)
31 \( 1 + 4.82T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 3.65T + 41T^{2} \)
43 \( 1 + 1.65iT - 43T^{2} \)
47 \( 1 - 5.17iT - 47T^{2} \)
53 \( 1 + 4.82iT - 53T^{2} \)
59 \( 1 + 11.6iT - 59T^{2} \)
61 \( 1 + 10.8iT - 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 - 12iT - 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + 3.65iT - 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−10.06151947573306679308713292583, −9.294963936247508416337851651317, −8.167477318869107018873175376798, −7.06277999723680409652632700400, −6.53669896637979029154368938996, −5.32349462335618342028690410072, −4.51803753802000874163196904978, −4.02643149161615767319243749354, −2.96097957775785336173588687997, −2.21830323465428303286520058551, 1.42182481179288454267752264335, 2.60237958747386684107664616286, 3.49164747946282631822532390516, 4.28091872104119901587234761275, 5.60540254135222778306499879626, 5.97998861060031927573964733672, 6.91051228397541307439956843436, 7.83702033635174386364952071648, 8.680633833851431841924344915536, 9.506160722708592013965544047785

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