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Properties

Label 2-1110-185.159-c1-0-30
Degree $2$
Conductor $1110$
Sign $0.993 - 0.111i$
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 + (0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (1.93 − 1.12i)5-s + 0.999i·6-s + (−3.60 − 2.08i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (1.93 + 1.11i)10-s + 5.62·11-s + (−0.866 + 0.499i)12-s + (2.80 − 4.85i)13-s − 4.16i·14-s + (2.23 − 0.00696i)15-s + (−0.5 − 0.866i)16-s + (−1.43 − 2.48i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.864 − 0.502i)5-s + 0.408i·6-s + (−1.36 − 0.786i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.613 + 0.351i)10-s + 1.69·11-s + (−0.249 + 0.144i)12-s + (0.777 − 1.34i)13-s − 1.11i·14-s + (0.577 − 0.00179i)15-s + (−0.125 − 0.216i)16-s + (−0.347 − 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.993 - 0.111i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.993 - 0.111i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.450053848\)
\(L(\frac12)\) \(\approx\) \(2.450053848\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-1.93 + 1.12i)T \)
37 \( 1 + (-2.01 - 5.73i)T \)
good7 \( 1 + (3.60 + 2.08i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.62T + 11T^{2} \)
13 \( 1 + (-2.80 + 4.85i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.43 + 2.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.16 + 2.98i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.24T + 23T^{2} \)
29 \( 1 - 9.90iT - 29T^{2} \)
31 \( 1 + 4.33iT - 31T^{2} \)
41 \( 1 + (0.470 - 0.815i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 4.29T + 43T^{2} \)
47 \( 1 - 0.365iT - 47T^{2} \)
53 \( 1 + (-3.87 + 2.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.52 + 5.49i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.20 - 1.27i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.92 + 1.10i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.08 - 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.5iT - 73T^{2} \)
79 \( 1 + (8.57 + 4.94i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.157 - 0.0907i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.38 - 4.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.96T + 97T^{2} \)
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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

−9.676240792878757188907595221645, −8.943193485987664570208758062442, −8.496956461044661483107078367734, −6.91031264895236887844400999047, −6.70716620464046683334217535783, −5.69153651964486213377080098929, −4.62493363730928571974500275429, −3.68866961575946902332876666907, −2.88531226229515667903300553100, −1.00587197081449440831934275632, 1.58406396156746397255463729550, 2.40983037140173440146960993674, 3.52086899950047916202787700748, 4.19547596161775609360841456495, 5.94160918931260437770414510350, 6.37281454883918694991483578337, 6.87815783943148634255369405128, 8.778286627176510759257994344015, 9.028220245560135055373668503291, 9.707545216108262947383696143084

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