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Properties

Label 2-1110-185.159-c1-0-1
Degree $2$
Conductor $1110$
Sign $-0.999 - 0.0393i$
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  + (0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (−1.62 − 1.53i)5-s − 0.999i·6-s + (2.75 + 1.58i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.516 − 2.17i)10-s − 3.88·11-s + (0.866 − 0.499i)12-s + (−0.379 + 0.657i)13-s + 3.17i·14-s + (0.640 + 2.14i)15-s + (−0.5 − 0.866i)16-s + (−1.69 − 2.93i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.726 − 0.686i)5-s − 0.408i·6-s + (1.04 + 0.600i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.163 − 0.687i)10-s − 1.16·11-s + (0.249 − 0.144i)12-s + (−0.105 + 0.182i)13-s + 0.849i·14-s + (0.165 + 0.553i)15-s + (−0.125 − 0.216i)16-s + (−0.410 − 0.711i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.999 - 0.0393i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.999 - 0.0393i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4142709576\)
\(L(\frac12)\) \(\approx\) \(0.4142709576\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (1.62 + 1.53i)T \)
37 \( 1 + (6.05 + 0.608i)T \)
good7 \( 1 + (-2.75 - 1.58i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.88T + 11T^{2} \)
13 \( 1 + (0.379 - 0.657i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.69 + 2.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.55 - 2.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.61T + 23T^{2} \)
29 \( 1 + 1.30iT - 29T^{2} \)
31 \( 1 - 6.12iT - 31T^{2} \)
41 \( 1 + (5.17 - 8.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 9.41T + 43T^{2} \)
47 \( 1 - 5.34iT - 47T^{2} \)
53 \( 1 + (-2.76 + 1.59i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.34 - 3.08i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.45 + 4.88i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.72 + 5.03i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.22 - 5.57i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.84iT - 73T^{2} \)
79 \( 1 + (-7.39 - 4.26i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.05 + 3.49i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.93 - 1.11i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.77T + 97T^{2} \)
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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

−10.29535003555146765515459010272, −9.201754411148769844121675357421, −8.115105158376927570598574672253, −7.978056903653656859635954863663, −7.00647511840774718018657091256, −5.81835640383508983037594119359, −5.03373946548564236120714200750, −4.66917242807013792250849967702, −3.22991138678626932073072390836, −1.70060865030750531708796902717, 0.16808795532936778959120368275, 1.93511485776089231457104902183, 3.23447929717361898771375257256, 4.15073495269648678600571128476, 4.93272133697772352195701129835, 5.78241306030015906335209776224, 7.00617807341693445219801855953, 7.76606735501591275776304761225, 8.508322500915710675173951004446, 9.885030508022656637294734614548

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