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Properties

Label 2-1110-5.4-c1-0-6
Degree $2$
Conductor $1110$
Sign $0.590 - 0.807i$
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  − i·2-s − i·3-s − 4-s + (1.80 + 1.32i)5-s − 6-s + 3.64i·7-s + i·8-s − 9-s + (1.32 − 1.80i)10-s − 4.60·11-s + i·12-s − 2.12i·13-s + 3.64·14-s + (1.32 − 1.80i)15-s + 16-s + 0.609i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.807 + 0.590i)5-s − 0.408·6-s + 1.37i·7-s + 0.353i·8-s − 0.333·9-s + (0.417 − 0.570i)10-s − 1.38·11-s + 0.288i·12-s − 0.589i·13-s + 0.972·14-s + (0.340 − 0.466i)15-s + 0.250·16-s + 0.147i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.590 - 0.807i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.590 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.106165603\)
\(L(\frac12)\) \(\approx\) \(1.106165603\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + iT \)
5 \( 1 + (-1.80 - 1.32i)T \)
37 \( 1 + iT \)
good7 \( 1 - 3.64iT - 7T^{2} \)
11 \( 1 + 4.60T + 11T^{2} \)
13 \( 1 + 2.12iT - 13T^{2} \)
17 \( 1 - 0.609iT - 17T^{2} \)
19 \( 1 + 3.48T + 19T^{2} \)
23 \( 1 - 8.76iT - 23T^{2} \)
29 \( 1 - 8.15T + 29T^{2} \)
31 \( 1 + 5.76T + 31T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 - 12.0iT - 43T^{2} \)
47 \( 1 - 1.60iT - 47T^{2} \)
53 \( 1 - 9.24iT - 53T^{2} \)
59 \( 1 - 3.67T + 59T^{2} \)
61 \( 1 - 1.45T + 61T^{2} \)
67 \( 1 + 3.75iT - 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + 6.76iT - 73T^{2} \)
79 \( 1 - 6.64T + 79T^{2} \)
83 \( 1 - 2.90iT - 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 16.5iT - 97T^{2} \)
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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

−10.04202871990548877030858195640, −9.261364626059781616927680800616, −8.386335104308975606474161796666, −7.66058082112068106476039983291, −6.43405368290077446793015124252, −5.61560312312749829324218823412, −5.08062773986615526895751105800, −3.21578305297437089325507711098, −2.57785956576013412961166683287, −1.71334344469056241331652760327, 0.46284078663440454033981612940, 2.29216865428278105128755647680, 3.82271190766000989911839054772, 4.73890053404317306995161131058, 5.23641998973541756427190477135, 6.46344334899647217177776852041, 7.05566722141842081941134730870, 8.336949386456094652358848080738, 8.626169651381033748286405972714, 9.916642254943977189459975011903

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