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Properties

Label 2-1110-37.36-c1-0-17
Degree $2$
Conductor $1110$
Sign $-0.784 + 0.620i$
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 − 3-s − 4-s + i·5-s + i·6-s − 7-s + i·8-s + 9-s + 10-s − 11-s + 12-s + 3.77i·13-s + i·14-s − i·15-s + 16-s + i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s − 0.377·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.04i·13-s + 0.267i·14-s − 0.258i·15-s + 0.250·16-s + 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.784 + 0.620i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.784 + 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6704018750\)
\(L(\frac12)\) \(\approx\) \(0.6704018750\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 - iT \)
37 \( 1 + (-4.77 + 3.77i)T \)
good7 \( 1 + T + 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 - 3.77iT - 13T^{2} \)
17 \( 1 - iT - 17T^{2} \)
19 \( 1 + 7.77iT - 19T^{2} \)
23 \( 1 + 7.77iT - 23T^{2} \)
29 \( 1 + 0.772iT - 29T^{2} \)
31 \( 1 + 0.772iT - 31T^{2} \)
41 \( 1 + 0.772T + 41T^{2} \)
43 \( 1 - 1.22iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + 7.54iT - 59T^{2} \)
61 \( 1 + 12.7iT - 61T^{2} \)
67 \( 1 - 9.54T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 0.227T + 73T^{2} \)
79 \( 1 + 11.5iT - 79T^{2} \)
83 \( 1 + 1.77T + 83T^{2} \)
89 \( 1 + 17.3iT - 89T^{2} \)
97 \( 1 + 10.7iT - 97T^{2} \)
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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

−9.617627854120319066019364220520, −8.995994241667549106645643137329, −7.907450019478383720211197708250, −6.76973644957434525780156071947, −6.30067219957520895055800718777, −4.96127408925626484358602737948, −4.30405368165470605622528409320, −3.05647866067484645448873503188, −2.06713684449809413188029944956, −0.34332338696476580850486373978, 1.29908672836102407920022994857, 3.19455780887877874026154625075, 4.22812394778794173437952286527, 5.44389636417750603702632394959, 5.70299252086931512555366913503, 6.77229265427398682634036357251, 7.77958133304127463421118610556, 8.220241552776647687487313725746, 9.483158107441210075601223831014, 9.957757155736101003210172650668

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