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Properties

Label 2-1155-385.384-c1-0-80
Degree $2$
Conductor $1155$
Sign $0.361 + 0.932i$
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  + 1.69·2-s − 3-s + 0.874·4-s + (2.22 + 0.217i)5-s − 1.69·6-s + (−0.698 − 2.55i)7-s − 1.90·8-s + 9-s + (3.77 + 0.368i)10-s + (3.25 − 0.658i)11-s − 0.874·12-s − 5.21i·13-s + (−1.18 − 4.32i)14-s + (−2.22 − 0.217i)15-s − 4.98·16-s − 3.07i·17-s + ⋯
L(s)  = 1  + 1.19·2-s − 0.577·3-s + 0.437·4-s + (0.995 + 0.0972i)5-s − 0.692·6-s + (−0.263 − 0.964i)7-s − 0.674·8-s + 0.333·9-s + (1.19 + 0.116i)10-s + (0.980 − 0.198i)11-s − 0.252·12-s − 1.44i·13-s + (−0.316 − 1.15i)14-s + (−0.574 − 0.0561i)15-s − 1.24·16-s − 0.746i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.361 + 0.932i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.361 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.507704398\)
\(L(\frac12)\) \(\approx\) \(2.507704398\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + (-2.22 - 0.217i)T \)
7 \( 1 + (0.698 + 2.55i)T \)
11 \( 1 + (-3.25 + 0.658i)T \)
good2 \( 1 - 1.69T + 2T^{2} \)
13 \( 1 + 5.21iT - 13T^{2} \)
17 \( 1 + 3.07iT - 17T^{2} \)
19 \( 1 + 4.27T + 19T^{2} \)
23 \( 1 - 6.74iT - 23T^{2} \)
29 \( 1 - 0.101iT - 29T^{2} \)
31 \( 1 + 6.52iT - 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 - 6.80T + 43T^{2} \)
47 \( 1 - 6.68T + 47T^{2} \)
53 \( 1 + 2.30iT - 53T^{2} \)
59 \( 1 - 7.18iT - 59T^{2} \)
61 \( 1 + 4.40T + 61T^{2} \)
67 \( 1 + 4.25iT - 67T^{2} \)
71 \( 1 + 5.83T + 71T^{2} \)
73 \( 1 + 8.09iT - 73T^{2} \)
79 \( 1 - 14.2iT - 79T^{2} \)
83 \( 1 - 7.09iT - 83T^{2} \)
89 \( 1 - 16.3iT - 89T^{2} \)
97 \( 1 + 2.98T + 97T^{2} \)
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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

−9.611531787121296719812596043259, −9.175392521546268516902908427530, −7.65766664921478321476116805798, −6.81761868483110226584204668102, −5.83538355675468566207715166838, −5.63026631061999630723396793097, −4.39115192140928173742191951586, −3.69265181330592123812652427430, −2.52702766398639978837091658439, −0.810383788209496322227718094973, 1.74193175519245445604651517131, 2.75977530059333940030278175556, 4.21368323414137603047421758420, 4.68955499298587525846255490510, 5.82334816771995937447502225602, 6.33136607598058268625118870353, 6.75730383080213394095097026190, 8.758229215409626149681245447842, 8.990998603982932849234185976568, 9.956988604044787246407669433037

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