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Properties

Label 2-1110-185.84-c1-0-14
Degree $2$
Conductor $1110$
Sign $0.497 - 0.867i$
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  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−1.75 − 1.39i)5-s − 0.999·6-s + (−0.903 + 0.521i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.820 − 2.07i)10-s + 0.877·11-s + (−0.866 − 0.499i)12-s + (3.82 − 2.20i)13-s − 1.04·14-s + (2.21 + 0.329i)15-s + (−0.5 + 0.866i)16-s + (4.44 + 2.56i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.783 − 0.622i)5-s − 0.408·6-s + (−0.341 + 0.197i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.259 − 0.657i)10-s + 0.264·11-s + (−0.249 − 0.144i)12-s + (1.05 − 0.611i)13-s − 0.278·14-s + (0.571 + 0.0849i)15-s + (−0.125 + 0.216i)16-s + (1.07 + 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.497 - 0.867i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.497 - 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.695707974\)
\(L(\frac12)\) \(\approx\) \(1.695707974\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (1.75 + 1.39i)T \)
37 \( 1 + (-0.0184 - 6.08i)T \)
good7 \( 1 + (0.903 - 0.521i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.877T + 11T^{2} \)
13 \( 1 + (-3.82 + 2.20i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.44 - 2.56i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.51 + 2.62i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.11iT - 23T^{2} \)
29 \( 1 - 5.96T + 29T^{2} \)
31 \( 1 + 1.73T + 31T^{2} \)
41 \( 1 + (-3.12 - 5.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 1.46iT - 43T^{2} \)
47 \( 1 + 3.00iT - 47T^{2} \)
53 \( 1 + (-11.1 - 6.44i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.32 + 7.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.80 - 3.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.30 + 1.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.582 - 1.00i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.73iT - 73T^{2} \)
79 \( 1 + (5.03 + 8.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.72 - 5.61i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.63 + 13.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.07iT - 97T^{2} \)
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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

−10.04431378328439883927640216532, −9.014847933276380977251803512153, −8.249950221208519378567343719920, −7.48289843502562465155111309210, −6.39519091013684361787456881149, −5.68041613028356939301847858566, −4.86287310107365836129132335419, −3.86505705818112738624113424398, −3.20817835469267760749042202132, −1.14002972158046632829711068624, 0.828805736352593545268239470767, 2.43317604633991441841315372063, 3.62420940564666123665757267936, 4.23127764832645854647288747400, 5.45159170635682986489396600982, 6.43770676031432852666401747230, 6.89376509436493766849341822947, 7.918630458623419687069637892423, 8.854212213372771208881119060144, 10.11801140272396091053885033849

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