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Properties

Label 2-1110-185.68-c1-0-28
Degree $2$
Conductor $1110$
Sign $-0.385 + 0.922i$
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 + (0.407 + 2.19i)5-s + (−0.707 + 0.707i)6-s + (1.64 − 1.64i)7-s − 8-s − 1.00i·9-s + (−0.407 − 2.19i)10-s − 5.09i·11-s + (0.707 − 0.707i)12-s − 5.90·13-s + (−1.64 + 1.64i)14-s + (1.84 + 1.26i)15-s + 16-s − 0.720i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (0.182 + 0.983i)5-s + (−0.288 + 0.288i)6-s + (0.621 − 0.621i)7-s − 0.353·8-s − 0.333i·9-s + (−0.128 − 0.695i)10-s − 1.53i·11-s + (0.204 − 0.204i)12-s − 1.63·13-s + (−0.439 + 0.439i)14-s + (0.475 + 0.327i)15-s + 0.250·16-s − 0.174i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.385 + 0.922i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.385 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9586188351\)
\(L(\frac12)\) \(\approx\) \(0.9586188351\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.407 - 2.19i)T \)
37 \( 1 + (-0.654 + 6.04i)T \)
good7 \( 1 + (-1.64 + 1.64i)T - 7iT^{2} \)
11 \( 1 + 5.09iT - 11T^{2} \)
13 \( 1 + 5.90T + 13T^{2} \)
17 \( 1 + 0.720iT - 17T^{2} \)
19 \( 1 + (5.20 + 5.20i)T + 19iT^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 + (2.09 - 2.09i)T - 29iT^{2} \)
31 \( 1 + (6.99 + 6.99i)T + 31iT^{2} \)
41 \( 1 - 6.58iT - 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + (-6.69 + 6.69i)T - 47iT^{2} \)
53 \( 1 + (-6.02 - 6.02i)T + 53iT^{2} \)
59 \( 1 + (1.31 + 1.31i)T + 59iT^{2} \)
61 \( 1 + (0.136 + 0.136i)T + 61iT^{2} \)
67 \( 1 + (8.82 + 8.82i)T + 67iT^{2} \)
71 \( 1 - 0.630T + 71T^{2} \)
73 \( 1 + (-0.800 + 0.800i)T - 73iT^{2} \)
79 \( 1 + (-1.35 - 1.35i)T + 79iT^{2} \)
83 \( 1 + (-0.0877 - 0.0877i)T + 83iT^{2} \)
89 \( 1 + (-0.651 + 0.651i)T - 89iT^{2} \)
97 \( 1 - 5.99iT - 97T^{2} \)
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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

−9.389937178538392832628040533198, −8.882507517605870692773366097408, −7.64451877676719996260136903731, −7.43594633946071841102691971870, −6.50545518567750126299261459436, −5.54190037028696917903106508493, −4.12943108740509159965369329879, −2.88450293166919523198950148721, −2.17187145525104539239424033564, −0.47079813340032225802360187704, 1.75937443936834171145777515322, 2.36864111341899811319778970542, 4.14143611090335033084564179009, 4.90815925171786714674970914922, 5.69846810959659081486839080543, 7.14275579976078922993134605517, 7.76334794184973925468380557024, 8.673033611088534783850726616597, 9.192054792091313074358049233831, 9.967530120715388639247133219535

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