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Properties

Label 2-1110-5.4-c1-0-24
Degree $2$
Conductor $1110$
Sign $0.894 + 0.447i$
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  + i·2-s − i·3-s − 4-s + (1 − 2i)5-s + 6-s + i·7-s − i·8-s − 9-s + (2 + i)10-s + 5·11-s + i·12-s − 14-s + (−2 − i)15-s + 16-s + i·17-s − i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.447 − 0.894i)5-s + 0.408·6-s + 0.377i·7-s − 0.353i·8-s − 0.333·9-s + (0.632 + 0.316i)10-s + 1.50·11-s + 0.288i·12-s − 0.267·14-s + (−0.516 − 0.258i)15-s + 0.250·16-s + 0.242i·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.734509122\)
\(L(\frac12)\) \(\approx\) \(1.734509122\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + iT \)
5 \( 1 + (-1 + 2i)T \)
37 \( 1 + iT \)
good7 \( 1 - iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
41 \( 1 + T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 5T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 + 8T + 89T^{2} \)
97 \( 1 + iT - 97T^{2} \)
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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

−9.461932428575362119272997614415, −8.742533599501486685955799282245, −8.346948234449979839643579756319, −7.13639854461668622546552540165, −6.40974912693474023525917083813, −5.72328564930003353726158881728, −4.75956057505067696990028294807, −3.78019112165209341356467054099, −2.12256468552869514027262606471, −0.895888343365722567060548398489, 1.35581878153588960545147587834, 2.71631083381843678212150511518, 3.63997426744686948500345781353, 4.38541092871708392793850025358, 5.60384256760759524094402033659, 6.50567474779619146881879836596, 7.34736670098169758450464632897, 8.554724625608482015802809910950, 9.432365744164382493535185519442, 9.898806548509306387902157028838

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