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Properties

Label 2-1110-37.27-c1-0-18
Degree $2$
Conductor $1110$
Sign $0.645 + 0.763i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s − 0.999i·6-s + (−0.839 − 1.45i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s − 3.11·11-s + (−0.499 + 0.866i)12-s + (1.90 − 1.09i)13-s + 1.67i·14-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s + (3.49 + 2.01i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s − 0.408i·6-s + (−0.317 − 0.549i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s − 0.316·10-s − 0.939·11-s + (−0.144 + 0.249i)12-s + (0.527 − 0.304i)13-s + 0.448i·14-s + (0.223 + 0.129i)15-s + (−0.125 + 0.216i)16-s + (0.848 + 0.489i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.645 + 0.763i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.645 + 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.291969562\)
\(L(\frac12)\) \(\approx\) \(1.291969562\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (-5.10 - 3.30i)T \)
good7 \( 1 + (0.839 + 1.45i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.11T + 11T^{2} \)
13 \( 1 + (-1.90 + 1.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.49 - 2.01i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.21 + 1.85i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.00iT - 23T^{2} \)
29 \( 1 + 5.04iT - 29T^{2} \)
31 \( 1 + 4.94iT - 31T^{2} \)
41 \( 1 + (-1.34 - 2.32i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 5.77iT - 43T^{2} \)
47 \( 1 - 5.05T + 47T^{2} \)
53 \( 1 + (6.38 - 11.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.36 + 2.52i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.3 + 5.97i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.18 - 3.78i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.36 + 2.35i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.06T + 73T^{2} \)
79 \( 1 + (-14.1 + 8.18i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.69 + 4.67i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.79 + 3.34i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.35iT - 97T^{2} \)
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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

−9.823557690311900372151166778919, −9.081934369798917094942515693836, −8.093393496041813225970289874930, −7.63418105656054241329791947297, −6.37055396474011021793610371680, −5.45184521759040019539818297786, −4.32508066476133824514398264731, −3.31175810389398949489377855455, −2.35638678516417943280556014481, −0.76560236535584033087310182388, 1.24657218836252751352684648911, 2.50574363663987810416939435459, 3.42655946512630589808923200559, 5.22292391905743338124730876578, 5.79383647732726362176372520521, 6.76630287645974386124862734271, 7.55754889888710974086340462654, 8.223879418186917440363881604071, 9.195572672903731868975064555505, 9.696814147638957739338473592217

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