Properties

Label 2-1110-37.26-c1-0-21
Degree $2$
Conductor $1110$
Sign $-0.967 + 0.251i$
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 + (2.09 − 3.62i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s − 5.42·11-s + (−0.499 − 0.866i)12-s + (2.00 − 3.46i)13-s − 4.18·14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (2.86 + 4.96i)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 + (0.790 − 1.36i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s − 1.63·11-s + (−0.144 − 0.249i)12-s + (0.555 − 0.961i)13-s − 1.11·14-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.694 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.967 + 0.251i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.967 + 0.251i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4090147672\)
\(L(\frac12)\) \(\approx\) \(0.4090147672\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.85 + 3.66i)T \)
good7 \( 1 + (-2.09 + 3.62i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 5.42T + 11T^{2} \)
13 \( 1 + (-2.00 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.86 - 4.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.631 - 1.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.55T + 23T^{2} \)
29 \( 1 - 0.557T + 29T^{2} \)
31 \( 1 + 5.22T + 31T^{2} \)
41 \( 1 + (-0.382 + 0.662i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 - 9.95T + 47T^{2} \)
53 \( 1 + (2.62 + 4.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.76 + 11.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.04 + 12.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.69 - 4.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.70 - 6.42i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 16.0T + 73T^{2} \)
79 \( 1 + (-3.35 + 5.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.71 + 2.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.45 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.137T + 97T^{2} \)
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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

−10.02890702196778401531785815536, −8.430983653631210690088084954657, −7.977811555023334861975424705905, −7.30724233319358291132837418694, −5.89728276903165426295718557322, −5.02021667888574954522630912138, −3.93462540533175984843476062906, −3.31799258226891378768506310736, −1.77336913306248169021338905229, −0.20713576853572433885579660203, 1.63333789726635098541674284946, 2.71932402457632544111073779366, 4.53782709780193027060834327428, 5.39530161158130471373403873214, 5.79840446432145144048795631965, 7.06249843369903257756262568476, 7.80769685098220621202171238100, 8.486064667156162274494903149543, 9.075125455219823858857293640205, 10.12561899289402140309080312689

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