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Properties

Label 2-1099-1099.1098-c0-0-4
Degree $2$
Conductor $1099$
Sign $-1$
Analytic cond. $0.548472$
Root an. cond. $0.740589$
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  − 1.41i·2-s + 1.41i·3-s − 1.00·4-s − 5-s + 2.00·6-s − 7-s − 1.00·9-s + 1.41i·10-s − 11-s − 1.41i·12-s − 1.41i·13-s + 1.41i·14-s − 1.41i·15-s − 0.999·16-s − 1.41i·17-s + 1.41i·18-s + ⋯
L(s)  = 1  − 1.41i·2-s + 1.41i·3-s − 1.00·4-s − 5-s + 2.00·6-s − 7-s − 1.00·9-s + 1.41i·10-s − 11-s − 1.41i·12-s − 1.41i·13-s + 1.41i·14-s − 1.41i·15-s − 0.999·16-s − 1.41i·17-s + 1.41i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1099 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1099 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1099\)    =    \(7 \cdot 157\)
Sign: $-1$
Analytic conductor: \(0.548472\)
Root analytic conductor: \(0.740589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1099} (1098, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1099,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2979770427\)
\(L(\frac12)\) \(\approx\) \(0.2979770427\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T \)
157 \( 1 - T \)
good2 \( 1 + 1.41iT - T^{2} \)
3 \( 1 - 1.41iT - T^{2} \)
5 \( 1 + T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−10.02697127879465102286653247363, −9.317730123173970563400941452994, −8.263320345692457528628780668144, −7.39968368005523691050150253004, −5.93298931425247340172289471092, −4.84992475220331956081163210841, −4.05683894913961140945146289398, −3.24586272782000393369981144265, −2.74764581226365098919858753969, −0.25379040125094437141265217263, 1.98831139872637300240300375714, 3.42465057177099296315430515807, 4.68319198866827924097166800081, 5.82888592592559224138222492326, 6.56221271945391349652557603056, 7.24243998820386445480635695261, 7.52162000766030121997088979968, 8.542538496249118108904771575075, 9.050300994705827572855263228799, 10.46829192724862959251865565560

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