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Properties

Label 2-1110-185.142-c1-0-2
Degree $2$
Conductor $1110$
Sign $0.830 - 0.556i$
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 + (−0.707 − 0.707i)3-s − 4-s + (−0.887 − 2.05i)5-s + (−0.707 + 0.707i)6-s + (−1.84 − 1.84i)7-s + i·8-s + 1.00i·9-s + (−2.05 + 0.887i)10-s + 2.12i·11-s + (0.707 + 0.707i)12-s + 5.07i·13-s + (−1.84 + 1.84i)14-s + (−0.823 + 2.07i)15-s + 16-s − 2.93·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (−0.397 − 0.917i)5-s + (−0.288 + 0.288i)6-s + (−0.697 − 0.697i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.648 + 0.280i)10-s + 0.639i·11-s + (0.204 + 0.204i)12-s + 1.40i·13-s + (−0.492 + 0.492i)14-s + (−0.212 + 0.536i)15-s + 0.250·16-s − 0.711·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.830 - 0.556i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.830 - 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3874885958\)
\(L(\frac12)\) \(\approx\) \(0.3874885958\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.887 + 2.05i)T \)
37 \( 1 + (5.46 + 2.66i)T \)
good7 \( 1 + (1.84 + 1.84i)T + 7iT^{2} \)
11 \( 1 - 2.12iT - 11T^{2} \)
13 \( 1 - 5.07iT - 13T^{2} \)
17 \( 1 + 2.93T + 17T^{2} \)
19 \( 1 + (2.28 + 2.28i)T + 19iT^{2} \)
23 \( 1 - 5.76iT - 23T^{2} \)
29 \( 1 + (-5.76 + 5.76i)T - 29iT^{2} \)
31 \( 1 + (-1.38 - 1.38i)T + 31iT^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 - 2.00iT - 43T^{2} \)
47 \( 1 + (-7.27 - 7.27i)T + 47iT^{2} \)
53 \( 1 + (0.286 - 0.286i)T - 53iT^{2} \)
59 \( 1 + (-8.25 - 8.25i)T + 59iT^{2} \)
61 \( 1 + (-2.44 - 2.44i)T + 61iT^{2} \)
67 \( 1 + (9.89 - 9.89i)T - 67iT^{2} \)
71 \( 1 - 1.69T + 71T^{2} \)
73 \( 1 + (-5.92 - 5.92i)T + 73iT^{2} \)
79 \( 1 + (5.73 + 5.73i)T + 79iT^{2} \)
83 \( 1 + (7.41 - 7.41i)T - 83iT^{2} \)
89 \( 1 + (11.9 - 11.9i)T - 89iT^{2} \)
97 \( 1 + 7.94T + 97T^{2} \)
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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

−9.906918570971096193234159354125, −9.205354692927187737789572523050, −8.481701864040425423590117691942, −7.30508329265178798978896156958, −6.73896598779751588634961146537, −5.52384182124640917012379272820, −4.37839918265468501440486002187, −4.02583649760473708160298605596, −2.37128199322287951337213737594, −1.18840991045606660790920163743, 0.20069394796315801051191386275, 2.75423825017637774567884624824, 3.51365720762610695061361972443, 4.69765847874273107103577457520, 5.75438495389529667366999163988, 6.34551999483115206130858309054, 7.03495057546071594560683827780, 8.279951564727496054501917660059, 8.660904330345204075550372313053, 9.926732196945060197313801146143

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