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Properties

Label 2-1155-77.76-c1-0-13
Degree $2$
Conductor $1155$
Sign $-0.964 + 0.262i$
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  + 2.40i·2-s − i·3-s − 3.78·4-s − i·5-s + 2.40·6-s + (−1.69 + 2.03i)7-s − 4.29i·8-s − 9-s + 2.40·10-s + (2.71 + 1.90i)11-s + 3.78i·12-s + 4.65·13-s + (−4.89 − 4.06i)14-s − 15-s + 2.76·16-s − 0.659·17-s + ⋯
L(s)  = 1  + 1.70i·2-s − 0.577i·3-s − 1.89·4-s − 0.447i·5-s + 0.982·6-s + (−0.638 + 0.769i)7-s − 1.51i·8-s − 0.333·9-s + 0.760·10-s + (0.818 + 0.574i)11-s + 1.09i·12-s + 1.28·13-s + (−1.30 − 1.08i)14-s − 0.258·15-s + 0.691·16-s − 0.159·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.964 + 0.262i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.964 + 0.262i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9829786042\)
\(L(\frac12)\) \(\approx\) \(0.9829786042\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 2.03i)T \)
11 \( 1 + (-2.71 - 1.90i)T \)
good2 \( 1 - 2.40iT - 2T^{2} \)
13 \( 1 - 4.65T + 13T^{2} \)
17 \( 1 + 0.659T + 17T^{2} \)
19 \( 1 - 0.0804T + 19T^{2} \)
23 \( 1 + 3.80T + 23T^{2} \)
29 \( 1 - 8.10iT - 29T^{2} \)
31 \( 1 - 3.95iT - 31T^{2} \)
37 \( 1 + 5.99T + 37T^{2} \)
41 \( 1 + 4.39T + 41T^{2} \)
43 \( 1 - 5.26iT - 43T^{2} \)
47 \( 1 - 7.38iT - 47T^{2} \)
53 \( 1 + 3.67T + 53T^{2} \)
59 \( 1 - 6.62iT - 59T^{2} \)
61 \( 1 + 14.5T + 61T^{2} \)
67 \( 1 + 6.70T + 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 + 6.71T + 73T^{2} \)
79 \( 1 - 4.24iT - 79T^{2} \)
83 \( 1 - 2.35T + 83T^{2} \)
89 \( 1 - 10.1iT - 89T^{2} \)
97 \( 1 + 4.04iT - 97T^{2} \)
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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

−9.743661655240950878260322056698, −8.861886269536568564620021306482, −8.653338633018490953508691408456, −7.64662842112878632642891759315, −6.73267527896575392959437582994, −6.26483727134970170352929406704, −5.52465885289145684226983204548, −4.55000063374298622326168078948, −3.37073671995060707314188684359, −1.54944442667164568404059444199, 0.44604893229305955766823729844, 1.86695619788672407727276642950, 3.25573457820779554584635450923, 3.71558044793375943839904523437, 4.39466616206848346044232007161, 5.88322751968125853670842324244, 6.67322632858200166521431140384, 8.085664666840553895323855559690, 8.977718967049569686931313153068, 9.642131352214773368795138781869

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