Properties

Label 2-1155-1155.1154-c0-0-8
Degree $2$
Conductor $1155$
Sign $0.866 + 0.5i$
Analytic cond. $0.576420$
Root an. cond. $0.759223$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·2-s + i·3-s − 1.99·4-s + (−0.866 − 0.5i)5-s − 1.73·6-s i·7-s − 1.73i·8-s − 9-s + (0.866 − 1.49i)10-s − 11-s − 1.99i·12-s i·13-s + 1.73·14-s + (0.5 − 0.866i)15-s + 0.999·16-s + ⋯
L(s)  = 1  + 1.73i·2-s + i·3-s − 1.99·4-s + (−0.866 − 0.5i)5-s − 1.73·6-s i·7-s − 1.73i·8-s − 9-s + (0.866 − 1.49i)10-s − 11-s − 1.99i·12-s i·13-s + 1.73·14-s + (0.5 − 0.866i)15-s + 0.999·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(0.576420\)
Root analytic conductor: \(0.759223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (1154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :0),\ 0.866 + 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06270380737\)
\(L(\frac12)\) \(\approx\) \(0.06270380737\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + iT \)
11 \( 1 + T \)
good2 \( 1 - 1.73iT - T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.73T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.73T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.73iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.716779984835265948294111284714, −8.759073303318826739932359933332, −8.036029158431686010515854727605, −7.73226514464078864908299341301, −6.59312752757852423803732211797, −5.65059145788940228130233170726, −4.76844259961484269743955896908, −4.32454306717423570288633879372, −3.25896897379239515047641494336, −0.05280534375367896970166015406, 1.99781546648243450477719369885, 2.47158519213021804123788461998, 3.53661093846183708486206301997, 4.55864405236426894536662985868, 5.74770087940335500721356019633, 6.81862783948448530626835031616, 7.83066427009607932193316082783, 8.661092456410526257751963938299, 9.188804961892295477213723660269, 10.46973282102024591446341778399

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