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Properties

Label 2-1110-37.10-c1-0-0
Degree $2$
Conductor $1110$
Sign $-0.227 - 0.973i$
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 + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·6-s + (1.43 + 2.48i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s − 5·11-s + (−0.499 + 0.866i)12-s + (0.5 + 0.866i)13-s + 2.87·14-s + (0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−3.93 + 6.81i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s − 0.408·6-s + (0.542 + 0.940i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + 0.316·10-s − 1.50·11-s + (−0.144 + 0.249i)12-s + (0.138 + 0.240i)13-s + 0.767·14-s + (0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.954 + 1.65i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.227 - 0.973i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.227 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4314546055\)
\(L(\frac12)\) \(\approx\) \(0.4314546055\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (5.5 + 2.59i)T \)
good7 \( 1 + (-1.43 - 2.48i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.93 - 6.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.87 + 6.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 3.87T + 29T^{2} \)
31 \( 1 + 7.74T + 31T^{2} \)
41 \( 1 + (-3.43 - 5.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 5.74T + 43T^{2} \)
47 \( 1 - 4.74T + 47T^{2} \)
53 \( 1 + (-3.43 + 5.95i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.563 - 0.976i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.93 + 12.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.30 - 7.46i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 12T + 73T^{2} \)
79 \( 1 + (-7.74 - 13.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.43 + 4.22i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.87 + 3.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−10.53692114433661094717478812422, −9.209450944810297986802880257362, −8.515737854766942713356608180607, −7.65522254502281583512792380058, −6.49979345940831128883111996484, −5.75679109810999361712380687226, −5.01980876404666399382892441322, −3.89313109923407137948511970329, −2.31790093745454089451763364125, −2.09675456517689442749644982989, 0.15720586943179854383510977451, 2.20489694261606872251258112094, 3.72472311161881320845239523502, 4.50340717743588221938106688197, 5.32230347756685828342026682348, 5.96300439519435644038236665341, 7.26413735868326665108743838013, 7.77658013477852514962561732297, 8.680615515796249402446336406569, 9.609235257526014294180228526572

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