Properties

Label 2-1110-37.11-c1-0-4
Degree $2$
Conductor $1110$
Sign $-0.607 - 0.794i$
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 + (0.435 − 0.754i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + 1.12·11-s + (0.499 + 0.866i)12-s + (−5.16 − 2.98i)13-s + 0.871i·14-s + (0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + (−1.62 + 0.935i)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 + (0.164 − 0.285i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s + 0.340·11-s + (0.144 + 0.249i)12-s + (−1.43 − 0.826i)13-s + 0.232i·14-s + (0.223 − 0.129i)15-s + (−0.125 − 0.216i)16-s + (−0.393 + 0.226i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.607 - 0.794i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.607 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6101300781\)
\(L(\frac12)\) \(\approx\) \(0.6101300781\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 + 5.69i)T \)
good7 \( 1 + (-0.435 + 0.754i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.12T + 11T^{2} \)
13 \( 1 + (5.16 + 2.98i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.62 - 0.935i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.64 - 3.25i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.22iT - 23T^{2} \)
29 \( 1 - 4.16iT - 29T^{2} \)
31 \( 1 - 5.15iT - 31T^{2} \)
41 \( 1 + (5.31 - 9.20i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 9.37iT - 43T^{2} \)
47 \( 1 - 7.61T + 47T^{2} \)
53 \( 1 + (-5.57 - 9.65i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.96 - 1.13i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.20 - 2.42i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.07 - 12.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.55 - 13.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.43T + 73T^{2} \)
79 \( 1 + (10.0 + 5.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.47 + 4.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.87 + 4.54i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.17iT - 97T^{2} \)
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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

−10.19783266494695660746611062120, −9.285693666167843219926418554418, −8.600665960319420228882194524229, −7.44778295594102888340571043461, −7.21923671114042725033150075767, −5.74133708941518536418287939822, −5.18145208288208141504721221073, −4.10443310060557116247424861935, −2.94573596534177495072231482021, −1.23386513455337320660632807960, 0.37430314838962755349150649676, 1.99226198316390518190973317740, 2.87077623673674260259585230644, 4.28373261331998305174595340380, 5.20684837834817776726403142594, 6.52516948731634458568506975628, 7.14637284670367843805681297849, 7.82971021706111740887895398585, 8.821841794152587132045536428937, 9.533664671609257592618610707242

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