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Properties

Label 2-1110-185.84-c1-0-27
Degree $2$
Conductor $1110$
Sign $-0.187 + 0.982i$
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 + (−2.14 + 0.629i)5-s − 0.999·6-s + (−1.82 + 1.05i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−2.17 − 0.527i)10-s + 1.56·11-s + (−0.866 − 0.499i)12-s + (−0.735 + 0.424i)13-s − 2.11·14-s + (1.54 − 1.61i)15-s + (−0.5 + 0.866i)16-s + (−5.36 − 3.09i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.959 + 0.281i)5-s − 0.408·6-s + (−0.691 + 0.399i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.687 − 0.166i)10-s + 0.472·11-s + (−0.249 − 0.144i)12-s + (−0.204 + 0.117i)13-s − 0.564·14-s + (0.398 − 0.417i)15-s + (−0.125 + 0.216i)16-s + (−1.30 − 0.751i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.187 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2104373612\)
\(L(\frac12)\) \(\approx\) \(0.2104373612\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 - 0.629i)T \)
37 \( 1 + (-3.54 + 4.94i)T \)
good7 \( 1 + (1.82 - 1.05i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.56T + 11T^{2} \)
13 \( 1 + (0.735 - 0.424i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.36 + 3.09i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.926 + 1.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.02iT - 23T^{2} \)
29 \( 1 - 2.74T + 29T^{2} \)
31 \( 1 + 5.35T + 31T^{2} \)
41 \( 1 + (1.71 + 2.97i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 2.80iT - 43T^{2} \)
47 \( 1 + 8.42iT - 47T^{2} \)
53 \( 1 + (7.19 + 4.15i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.267 + 0.463i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.39 + 9.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0321 + 0.0185i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.11 + 8.85i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.96iT - 73T^{2} \)
79 \( 1 + (-4.21 - 7.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.938 + 0.542i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.64 - 8.04i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.7iT - 97T^{2} \)
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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

−9.475903425775874264829010847205, −8.846230705142210495922638369934, −7.76561755018658241848166173233, −6.75237204652071540613191779792, −6.47312726631637458456598994075, −5.19870065100243725777855991986, −4.40241612149001588562628430763, −3.57403920317894274183162452542, −2.49844907006873483637567377082, −0.080442888690161024872224874498, 1.46681176443675011221001986161, 3.03308189460316160138592569855, 4.05485974129671507811831110910, 4.64025285977331483535073449419, 5.88586894628987325951671931604, 6.63892750643284037553091191821, 7.37943013079336138222619920368, 8.385436872661575243440210221522, 9.339611699394752542140869430864, 10.33034510092430956663582261531

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