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Properties

Label 2-1155-77.76-c1-0-31
Degree $2$
Conductor $1155$
Sign $0.871 - 0.490i$
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  + 0.517i·2-s − i·3-s + 1.73·4-s + i·5-s + 0.517·6-s + (2.63 − 0.189i)7-s + 1.93i·8-s − 9-s − 0.517·10-s + (−3 + 1.41i)11-s − 1.73i·12-s + 4.24·13-s + (0.0980 + 1.36i)14-s + 15-s + 2.46·16-s + 2.44·17-s + ⋯
L(s)  = 1  + 0.366i·2-s − 0.577i·3-s + 0.866·4-s + 0.447i·5-s + 0.211·6-s + (0.997 − 0.0716i)7-s + 0.683i·8-s − 0.333·9-s − 0.163·10-s + (−0.904 + 0.426i)11-s − 0.499i·12-s + 1.17·13-s + (0.0262 + 0.365i)14-s + 0.258·15-s + 0.616·16-s + 0.594·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.871 - 0.490i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.871 - 0.490i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.293097059\)
\(L(\frac12)\) \(\approx\) \(2.293097059\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 - iT \)
7 \( 1 + (-2.63 + 0.189i)T \)
11 \( 1 + (3 - 1.41i)T \)
good2 \( 1 - 0.517iT - 2T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 - 2.44T + 17T^{2} \)
19 \( 1 - 1.03T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 1.03iT - 29T^{2} \)
31 \( 1 - 2.92iT - 31T^{2} \)
37 \( 1 + 0.535T + 37T^{2} \)
41 \( 1 - 6.69T + 41T^{2} \)
43 \( 1 - 2.44iT - 43T^{2} \)
47 \( 1 + 2.92iT - 47T^{2} \)
53 \( 1 + 3.46T + 53T^{2} \)
59 \( 1 + 12.9iT - 59T^{2} \)
61 \( 1 + 4.89T + 61T^{2} \)
67 \( 1 + 5.46T + 67T^{2} \)
71 \( 1 - 6.92T + 71T^{2} \)
73 \( 1 + 0.656T + 73T^{2} \)
79 \( 1 + 3.86iT - 79T^{2} \)
83 \( 1 - 1.41T + 83T^{2} \)
89 \( 1 + 4.92iT - 89T^{2} \)
97 \( 1 - 4.92iT - 97T^{2} \)
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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

−10.08335105344579291989134144786, −8.706784338157035549540419996627, −7.82348560382796709433859941141, −7.60604011643132907307695008000, −6.54278062482656326235443753699, −5.83596527940718132066496087348, −4.96859349686474719569485060844, −3.49972787533585385789511917688, −2.37894560704145831422315347377, −1.44476854162394081452273797114, 1.14805955310061395732974199869, 2.36331837421152666844938495890, 3.47552653823499652681979587580, 4.42261044718437761568336473139, 5.57681672188928364856654419671, 6.05727130164354052414904241740, 7.52577167342049437622318738885, 8.071289676527399896814547328704, 8.898405127629227581078996736293, 9.947084024417875847734536090614

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