Properties

Label 2-1110-37.10-c1-0-19
Degree $2$
Conductor $1110$
Sign $-0.231 + 0.972i$
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 + (−0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s − 2·11-s + (−0.499 + 0.866i)12-s + (0.5 + 0.866i)13-s + 0.999·14-s + (0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)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 + (−0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s − 0.316·10-s − 0.603·11-s + (−0.144 + 0.249i)12-s + (0.138 + 0.240i)13-s + 0.267·14-s + (0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.242 − 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.231 + 0.972i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.231 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5651267782\)
\(L(\frac12)\) \(\approx\) \(0.5651267782\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.76 + 4.78i)T \)
good7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.76 + 4.78i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 5.52T + 29T^{2} \)
31 \( 1 + 2.52T + 31T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.52 + 7.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.260 + 0.451i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.239 - 0.415i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.52T + 73T^{2} \)
79 \( 1 + (1.26 + 2.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.23 + 3.87i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.52 + 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.52T + 97T^{2} \)
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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

−9.592196004880319674960385645199, −8.634273178213061539872369856757, −7.87758550025610728433955749827, −6.89475143073441270285859757783, −6.59416362851541573609678281181, −5.46293356998470165968778978431, −4.67571292449856941092535622557, −3.20808306918801663078071290235, −1.92728931955661281266567042587, −0.29108407088561789956321628625, 1.48497929366756751508745332580, 2.81499334434102648868542469132, 3.82729047466218231111988019509, 4.86674544262026924553633095874, 5.69927593445464258495301215574, 6.63471480118783950062696227810, 8.013236192130564381053978538424, 8.498753944060611820911770812890, 9.417265629494266173976187489947, 10.27504376082615526100892068597

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