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Properties

Label 2-1110-37.26-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 + (−2.43 + 4.22i)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 − 4.87·14-s + (0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.0635 − 0.109i)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 + (−0.920 + 1.59i)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 − 1.30·14-s + (0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.0154 − 0.0266i)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: $\chi_{1110} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.227 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1624885785\)
\(L(\frac12)\) \(\approx\) \(0.1624885785\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.43 - 4.22i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.0635 + 0.109i)T + (-8.5 + 14.7i)T^{2} \)
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 \( 1 + (0.436 - 0.756i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 9.74T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + (0.436 + 0.756i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.5 + 4.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.43 + 7.68i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.06 - 5.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.30 - 12.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 12T + 73T^{2} \)
79 \( 1 + (7.74 - 13.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.43 + 2.48i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.87 + 10.1i)T + (-44.5 + 77.0i)T^{2} \)
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

−9.995378838609540376229036673094, −9.758287997268664810722113389311, −8.585342049036459645374296501732, −8.232517167802057760872728366829, −6.82386246257520103460474509198, −6.06692044118994456916770043462, −5.24681201054180264290857008369, −4.85993832684766740526328212768, −3.23289390852232426430288888995, −2.54765684196726120504344510034, 0.06318243445680712428983334028, 1.54573221055314416946477809069, 2.93370613615104252888068511185, 3.71641186803508009783730983885, 4.83857505945172116988990650602, 5.92392035513148371797021592454, 6.62660683047404322503457119618, 7.56412890923132422094579560378, 8.209671298401209136122332057423, 9.864860136954390799522252870039

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