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Properties

Label 2-1110-185.84-c1-0-38
Degree $2$
Conductor $1110$
Sign $0.926 + 0.377i$
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 + (1.56 − 1.59i)5-s + 0.999·6-s + (2.76 − 1.59i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (2.15 − 0.597i)10-s − 4.26·11-s + (0.866 + 0.499i)12-s + (2.86 − 1.65i)13-s + 3.19·14-s + (0.559 − 2.16i)15-s + (−0.5 + 0.866i)16-s + (−0.167 − 0.0969i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.700 − 0.713i)5-s + 0.408·6-s + (1.04 − 0.603i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.681 − 0.188i)10-s − 1.28·11-s + (0.249 + 0.144i)12-s + (0.794 − 0.458i)13-s + 0.853·14-s + (0.144 − 0.558i)15-s + (−0.125 + 0.216i)16-s + (−0.0407 − 0.0235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.926 + 0.377i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.926 + 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.288303730\)
\(L(\frac12)\) \(\approx\) \(3.288303730\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.56 + 1.59i)T \)
37 \( 1 + (1.46 + 5.90i)T \)
good7 \( 1 + (-2.76 + 1.59i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.26T + 11T^{2} \)
13 \( 1 + (-2.86 + 1.65i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.167 + 0.0969i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.557 - 0.964i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.44iT - 23T^{2} \)
29 \( 1 + 4.19T + 29T^{2} \)
31 \( 1 - 1.04T + 31T^{2} \)
41 \( 1 + (-2.83 - 4.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 0.530iT - 43T^{2} \)
47 \( 1 - 12.6iT - 47T^{2} \)
53 \( 1 + (6.53 + 3.77i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.27 + 5.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.65 - 4.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.40 - 2.54i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.82 - 3.16i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.24iT - 73T^{2} \)
79 \( 1 + (-2.12 - 3.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-14.4 - 8.32i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.09 + 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.1iT - 97T^{2} \)
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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

−9.714040010229549971325941579079, −8.755411586352386077394981446366, −7.86927759460540376731532798556, −7.63692446764837742390601726535, −6.25565141094683333078433903143, −5.42917677477920061889073296080, −4.74625667016212401729766512236, −3.67129366696157896264264422316, −2.40714850658663323355448202351, −1.28201415284797932428140652199, 1.83216463122557336375237832020, 2.52640716262538515907323425934, 3.53818072423732850206851681496, 4.78156219810675184589224175654, 5.43326489987415170820007132771, 6.34534496461002159029345626201, 7.41512195959547117814306373059, 8.339255192678135388130824770891, 9.107367793494416006476862871143, 10.11606067942549893107091955070

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