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Properties

Label 2-1110-37.26-c1-0-22
Degree $2$
Conductor $1110$
Sign $0.729 + 0.683i$
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 + (1.5 − 2.59i)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 + (2.5 − 4.33i)13-s + 3·14-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 + (0.566 − 0.981i)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 + (0.693 − 1.20i)13-s + 0.801·14-s + (0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.242 − 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.729 + 0.683i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.729 + 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.871962756\)
\(L(\frac12)\) \(\approx\) \(1.871962756\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 6.06i)T \)
good7 \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
41 \( 1 + (-4 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4 - 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−9.790350235133806941771929506128, −8.441415028354372868602428960285, −7.981503235449848832805616885141, −7.31977588978599633885090909763, −6.52719516192561633612746474034, −5.54917928030929152353684108601, −4.55324535471431792469270831168, −3.55765471115212399688304960601, −2.54594265830483708551903119224, −0.73359836761843542586620506714, 1.63577135326902415603914677464, 2.59423676419490241974096623353, 3.89801382029029491995489873891, 4.50194207772694664740192597835, 5.55751317488754659758590245493, 6.21037750487360919173461094893, 7.82308230809700285648835024222, 8.439372116882129520433634442255, 9.170583782538535838100764164529, 9.962461631172528544532136633171

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