Properties

Label 2-1110-185.68-c1-0-32
Degree $2$
Conductor $1110$
Sign $-0.998 - 0.0623i$
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  − 2-s + (0.707 − 0.707i)3-s + 4-s + (−0.891 − 2.05i)5-s + (−0.707 + 0.707i)6-s + (−1.56 + 1.56i)7-s − 8-s − 1.00i·9-s + (0.891 + 2.05i)10-s − 3.92i·11-s + (0.707 − 0.707i)12-s − 0.319·13-s + (1.56 − 1.56i)14-s + (−2.08 − 0.819i)15-s + 16-s − 1.25i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (−0.398 − 0.917i)5-s + (−0.288 + 0.288i)6-s + (−0.591 + 0.591i)7-s − 0.353·8-s − 0.333i·9-s + (0.282 + 0.648i)10-s − 1.18i·11-s + (0.204 − 0.204i)12-s − 0.0885·13-s + (0.418 − 0.418i)14-s + (−0.537 − 0.211i)15-s + 0.250·16-s − 0.304i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.998 - 0.0623i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.998 - 0.0623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4477014545\)
\(L(\frac12)\) \(\approx\) \(0.4477014545\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.891 + 2.05i)T \)
37 \( 1 + (6.08 + 0.0100i)T \)
good7 \( 1 + (1.56 - 1.56i)T - 7iT^{2} \)
11 \( 1 + 3.92iT - 11T^{2} \)
13 \( 1 + 0.319T + 13T^{2} \)
17 \( 1 + 1.25iT - 17T^{2} \)
19 \( 1 + (-2.17 - 2.17i)T + 19iT^{2} \)
23 \( 1 + 1.15T + 23T^{2} \)
29 \( 1 + (1.22 - 1.22i)T - 29iT^{2} \)
31 \( 1 + (6.78 + 6.78i)T + 31iT^{2} \)
41 \( 1 + 3.92iT - 41T^{2} \)
43 \( 1 + 1.87T + 43T^{2} \)
47 \( 1 + (4.88 - 4.88i)T - 47iT^{2} \)
53 \( 1 + (3.03 + 3.03i)T + 53iT^{2} \)
59 \( 1 + (-4.70 - 4.70i)T + 59iT^{2} \)
61 \( 1 + (5.56 + 5.56i)T + 61iT^{2} \)
67 \( 1 + (-6.88 - 6.88i)T + 67iT^{2} \)
71 \( 1 + 3.29T + 71T^{2} \)
73 \( 1 + (8.53 - 8.53i)T - 73iT^{2} \)
79 \( 1 + (3.59 + 3.59i)T + 79iT^{2} \)
83 \( 1 + (1.13 + 1.13i)T + 83iT^{2} \)
89 \( 1 + (-1.33 + 1.33i)T - 89iT^{2} \)
97 \( 1 + 4.01iT - 97T^{2} \)
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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

−9.209839182982906263644242451354, −8.668255787349911739888263585672, −7.966458219159052913579432746777, −7.18886682128206650390160300069, −6.03869547651018991589989797079, −5.39901105845528228085057612196, −3.85354718708823515619239049716, −2.95035793655674615849535379096, −1.60474971943282737160485155821, −0.22815743327069628868051819372, 1.89934969479038953709704427807, 3.10514569833017167960609337408, 3.84889240197311568960808528160, 5.04247385244187540299714688176, 6.46359096114049168513422417858, 7.13868416893669774493831382164, 7.66375649515146420620853276957, 8.711852537418441285578396962314, 9.610853848805360717332550911901, 10.17010483027956954713021140045

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