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Properties

Label 2-1110-185.64-c1-0-28
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: $\chi_{1110} (619, \cdot )$
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

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

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