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Properties

Label 2-1110-185.43-c1-0-15
Degree $2$
Conductor $1110$
Sign $0.0844 - 0.996i$
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 + (−0.707 + 0.707i)3-s − 4-s + (−1.16 + 1.91i)5-s + (−0.707 − 0.707i)6-s + (2.90 − 2.90i)7-s − i·8-s − 1.00i·9-s + (−1.91 − 1.16i)10-s − 2.74i·11-s + (0.707 − 0.707i)12-s + 5.21i·13-s + (2.90 + 2.90i)14-s + (−0.529 − 2.17i)15-s + 16-s + 4.16·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (−0.519 + 0.854i)5-s + (−0.288 − 0.288i)6-s + (1.09 − 1.09i)7-s − 0.353i·8-s − 0.333i·9-s + (−0.604 − 0.367i)10-s − 0.828i·11-s + (0.204 − 0.204i)12-s + 1.44i·13-s + (0.776 + 0.776i)14-s + (−0.136 − 0.560i)15-s + 0.250·16-s + 1.00·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0844 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0844 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.0844 - 0.996i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.0844 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.397114631\)
\(L(\frac12)\) \(\approx\) \(1.397114631\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.16 - 1.91i)T \)
37 \( 1 + (-5.41 - 2.77i)T \)
good7 \( 1 + (-2.90 + 2.90i)T - 7iT^{2} \)
11 \( 1 + 2.74iT - 11T^{2} \)
13 \( 1 - 5.21iT - 13T^{2} \)
17 \( 1 - 4.16T + 17T^{2} \)
19 \( 1 + (-1.87 + 1.87i)T - 19iT^{2} \)
23 \( 1 + 0.941iT - 23T^{2} \)
29 \( 1 + (-4.28 - 4.28i)T + 29iT^{2} \)
31 \( 1 + (-5.51 + 5.51i)T - 31iT^{2} \)
41 \( 1 - 7.59iT - 41T^{2} \)
43 \( 1 + 6.23iT - 43T^{2} \)
47 \( 1 + (6.04 - 6.04i)T - 47iT^{2} \)
53 \( 1 + (-8.31 - 8.31i)T + 53iT^{2} \)
59 \( 1 + (1.92 - 1.92i)T - 59iT^{2} \)
61 \( 1 + (8.68 - 8.68i)T - 61iT^{2} \)
67 \( 1 + (5.51 + 5.51i)T + 67iT^{2} \)
71 \( 1 + 8.51T + 71T^{2} \)
73 \( 1 + (-2.98 + 2.98i)T - 73iT^{2} \)
79 \( 1 + (1.46 - 1.46i)T - 79iT^{2} \)
83 \( 1 + (-7.26 - 7.26i)T + 83iT^{2} \)
89 \( 1 + (-1.62 - 1.62i)T + 89iT^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−10.16747123693817671089540199147, −9.160182469818046645419301768212, −8.120419619634668434670653778198, −7.53925810905133214338484924819, −6.72876577699883002821834819579, −5.96550523466965187042330771448, −4.67210838348481099688806857296, −4.23081439595321131755166854199, −3.09443142659048198126938543370, −1.05376700583469063483013291921, 0.906056000536543057091347277175, 1.97362357810556261055155438299, 3.22522858908903772409585011706, 4.63023668531570095460102899987, 5.19784198029021608465137529127, 5.89714159072595191409235984811, 7.56033919293369304055329043133, 8.070639338404410998004707249554, 8.697217059954156963651240707983, 9.795501964932726263294880150183

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