Properties

Label 2-1110-111.68-c1-0-16
Degree $2$
Conductor $1110$
Sign $-0.0872 - 0.996i$
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  + (−0.707 + 0.707i)2-s + (1.41 + i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (−1.70 + 0.292i)6-s + 4·7-s + (0.707 + 0.707i)8-s + (1.00 + 2.82i)9-s + 1.00·10-s − 4.24·11-s + (1.00 − 1.41i)12-s + (−1 + i)13-s + (−2.82 + 2.82i)14-s + (−0.292 − 1.70i)15-s − 1.00·16-s + (1.41 + 1.41i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.816 + 0.577i)3-s − 0.500i·4-s + (−0.316 − 0.316i)5-s + (−0.696 + 0.119i)6-s + 1.51·7-s + (0.250 + 0.250i)8-s + (0.333 + 0.942i)9-s + 0.316·10-s − 1.27·11-s + (0.288 − 0.408i)12-s + (−0.277 + 0.277i)13-s + (−0.755 + 0.755i)14-s + (−0.0756 − 0.440i)15-s − 0.250·16-s + (0.342 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0872 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0872 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.0872 - 0.996i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.0872 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.689750394\)
\(L(\frac12)\) \(\approx\) \(1.689750394\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-6 - i)T \)
good7 \( 1 - 4T + 7T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 + (1 - i)T - 13iT^{2} \)
17 \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \)
19 \( 1 + (2 - 2i)T - 19iT^{2} \)
23 \( 1 + (-4.24 - 4.24i)T + 23iT^{2} \)
29 \( 1 - 29iT^{2} \)
31 \( 1 + (-5 - 5i)T + 31iT^{2} \)
41 \( 1 + 5.65T + 41T^{2} \)
43 \( 1 + (-9 + 9i)T - 43iT^{2} \)
47 \( 1 - 1.41iT - 47T^{2} \)
53 \( 1 - 2.82iT - 53T^{2} \)
59 \( 1 + (-2.82 - 2.82i)T + 59iT^{2} \)
61 \( 1 + (-4 - 4i)T + 61iT^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + (3 - 3i)T - 79iT^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 + (-12.7 + 12.7i)T - 89iT^{2} \)
97 \( 1 + (12 - 12i)T - 97iT^{2} \)
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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

−10.01557331369368587447359057125, −8.975582050782904072151136238992, −8.342828308349547007918214724953, −7.84081320725058623279873879045, −7.19161426308771528526955445868, −5.53208497836927928756840890794, −4.95229467080386844360094903867, −4.11857235334476121743580824559, −2.67021157358878932183785255397, −1.50009799207778545486043385307, 0.861569195168630739258680216513, 2.30196824802098305971659635939, 2.82989870082660568266227677528, 4.24498609967899558218797644174, 5.12421791754311686281628464165, 6.58832818291790810094777898682, 7.67475714634556260722835862912, 7.913654311374382653214691632121, 8.603085466394418220448473421810, 9.579064423965869101406996895641

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