Properties

Label 2-1110-111.80-c1-0-8
Degree $2$
Conductor $1110$
Sign $-0.799 - 0.601i$
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  + (−0.707 − 0.707i)2-s + (0.878 + 1.49i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (0.434 − 1.67i)6-s + 2.38·7-s + (0.707 − 0.707i)8-s + (−1.45 + 2.62i)9-s + 1.00·10-s − 5.76·11-s + (−1.49 + 0.878i)12-s + (−1.58 − 1.58i)13-s + (−1.68 − 1.68i)14-s + (−1.67 − 0.434i)15-s − 1.00·16-s + (0.766 − 0.766i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.507 + 0.861i)3-s + 0.500i·4-s + (−0.316 + 0.316i)5-s + (0.177 − 0.684i)6-s + 0.901·7-s + (0.250 − 0.250i)8-s + (−0.485 + 0.874i)9-s + 0.316·10-s − 1.73·11-s + (−0.430 + 0.253i)12-s + (−0.438 − 0.438i)13-s + (−0.450 − 0.450i)14-s + (−0.432 − 0.112i)15-s − 0.250·16-s + (0.186 − 0.186i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.799 - 0.601i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.799 - 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7532162905\)
\(L(\frac12)\) \(\approx\) \(0.7532162905\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-0.878 - 1.49i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (-5.79 - 1.85i)T \)
good7 \( 1 - 2.38T + 7T^{2} \)
11 \( 1 + 5.76T + 11T^{2} \)
13 \( 1 + (1.58 + 1.58i)T + 13iT^{2} \)
17 \( 1 + (-0.766 + 0.766i)T - 17iT^{2} \)
19 \( 1 + (-4.27 - 4.27i)T + 19iT^{2} \)
23 \( 1 + (5.87 - 5.87i)T - 23iT^{2} \)
29 \( 1 + (-0.666 - 0.666i)T + 29iT^{2} \)
31 \( 1 + (4.93 - 4.93i)T - 31iT^{2} \)
41 \( 1 + 2.85T + 41T^{2} \)
43 \( 1 + (6.47 + 6.47i)T + 43iT^{2} \)
47 \( 1 + 8.44iT - 47T^{2} \)
53 \( 1 - 7.21iT - 53T^{2} \)
59 \( 1 + (-5.63 + 5.63i)T - 59iT^{2} \)
61 \( 1 + (8.82 - 8.82i)T - 61iT^{2} \)
67 \( 1 - 2.96iT - 67T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 - 1.48iT - 73T^{2} \)
79 \( 1 + (3.85 + 3.85i)T + 79iT^{2} \)
83 \( 1 - 16.2iT - 83T^{2} \)
89 \( 1 + (-3.88 - 3.88i)T + 89iT^{2} \)
97 \( 1 + (-11.4 - 11.4i)T + 97iT^{2} \)
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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

−10.25115727543526653061429836720, −9.571151688775311793327007920298, −8.435107736574074398966749316088, −7.84325637196183736375624387644, −7.46939788975830235808397524467, −5.51769984633227275510893436812, −5.01944247915306777389984711492, −3.75469521542533265467642108122, −2.97345907348067394745903098814, −1.90218072171175919411337558829, 0.34507164394895710968578019081, 1.85995617732533763585448803659, 2.82878121688772365131567976172, 4.48113701250068291435025823076, 5.31241221887432088098444532163, 6.31233757781975323226309790403, 7.43245446838036018283738776334, 7.84955399024199043798953268464, 8.362335024966250875974922499951, 9.306560559158743708093189945159

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