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Properties

Label 2-1110-185.174-c1-0-33
Degree $2$
Conductor $1110$
Sign $-0.979 - 0.200i$
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 + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.808 − 2.08i)5-s + 0.999·6-s + (−3.36 − 1.94i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (0.342 + 2.20i)10-s − 0.297·11-s + (−0.866 + 0.499i)12-s + (0.273 + 0.157i)13-s + 3.88·14-s + (−1.74 + 1.40i)15-s + (−0.5 − 0.866i)16-s + (0.766 − 0.442i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.361 − 0.932i)5-s + 0.408·6-s + (−1.27 − 0.734i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.108 + 0.698i)10-s − 0.0896·11-s + (−0.249 + 0.144i)12-s + (0.0758 + 0.0438i)13-s + 1.03·14-s + (−0.449 + 0.361i)15-s + (−0.125 − 0.216i)16-s + (0.185 − 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.979 - 0.200i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.979 - 0.200i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2804451051\)
\(L(\frac12)\) \(\approx\) \(0.2804451051\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.808 + 2.08i)T \)
37 \( 1 + (-1.88 - 5.78i)T \)
good7 \( 1 + (3.36 + 1.94i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.297T + 11T^{2} \)
13 \( 1 + (-0.273 - 0.157i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.766 + 0.442i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.55 + 2.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.33iT - 23T^{2} \)
29 \( 1 + 4.88T + 29T^{2} \)
31 \( 1 - 1.38T + 31T^{2} \)
41 \( 1 + (6.17 - 10.7i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 7.34iT - 43T^{2} \)
47 \( 1 + 5.14iT - 47T^{2} \)
53 \( 1 + (4.71 - 2.72i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.37 + 11.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.26 - 2.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.512 + 0.295i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.61 - 4.53i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 14.4iT - 73T^{2} \)
79 \( 1 + (-1.22 + 2.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.61 - 3.24i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.65 - 4.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.87iT - 97T^{2} \)
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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

−9.618156279705469945670061271544, −8.567743163770058106430676776156, −7.78802999195206902037924205948, −6.69562659905340731075245375308, −6.33892772281091864991294065987, −5.24193722313923885410590058678, −4.34841340615996078527495220268, −2.87600170159116072435887266123, −1.28720719922025569915976085260, −0.16894031935217961992052909277, 1.90143704416594903073993622337, 3.12350885951280594448279941236, 3.72638413700293209999177920997, 5.54506550112475606940756616257, 6.00745508547773507895365693081, 6.98621533153247742850334136376, 7.69797623688694225101910560402, 9.105582827195389873143485971259, 9.484458094714763081678630056878, 10.26995039043561500471091334364

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