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Properties

Label 2-1110-111.80-c1-0-9
Degree $2$
Conductor $1110$
Sign $-0.968 + 0.249i$
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  + (0.707 + 0.707i)2-s + (1.41 + i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (0.292 + 1.70i)6-s − 4·7-s + (−0.707 + 0.707i)8-s + (1.00 + 2.82i)9-s − 1.00·10-s − 1.41·11-s + (−1.00 + 1.41i)12-s + (−3 − 3i)13-s + (−2.82 − 2.82i)14-s + (−1.70 + 0.292i)15-s − 1.00·16-s + (−1.41 + 1.41i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.816 + 0.577i)3-s + 0.500i·4-s + (−0.316 + 0.316i)5-s + (0.119 + 0.696i)6-s − 1.51·7-s + (−0.250 + 0.250i)8-s + (0.333 + 0.942i)9-s − 0.316·10-s − 0.426·11-s + (−0.288 + 0.408i)12-s + (−0.832 − 0.832i)13-s + (−0.755 − 0.755i)14-s + (−0.440 + 0.0756i)15-s − 0.250·16-s + (−0.342 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.968 + 0.249i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.968 + 0.249i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.244228083\)
\(L(\frac12)\) \(\approx\) \(1.244228083\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-1.41 - i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (6 - i)T \)
good7 \( 1 + 4T + 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (1.41 - 1.41i)T - 17iT^{2} \)
19 \( 1 + (-4 - 4i)T + 19iT^{2} \)
23 \( 1 + (1.41 - 1.41i)T - 23iT^{2} \)
29 \( 1 + (2.82 + 2.82i)T + 29iT^{2} \)
31 \( 1 + (3 - 3i)T - 31iT^{2} \)
41 \( 1 - 5.65T + 41T^{2} \)
43 \( 1 + (-3 - 3i)T + 43iT^{2} \)
47 \( 1 + 1.41iT - 47T^{2} \)
53 \( 1 - 2.82iT - 53T^{2} \)
59 \( 1 + (-8.48 + 8.48i)T - 59iT^{2} \)
61 \( 1 + (6 - 6i)T - 61iT^{2} \)
67 \( 1 - 6iT - 67T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + (-5 - 5i)T + 79iT^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 + (-1.41 - 1.41i)T + 89iT^{2} \)
97 \( 1 + (-6 - 6i)T + 97iT^{2} \)
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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

−10.02488620447908709534015211166, −9.594279500516419916423878344041, −8.556760046699948474979900510603, −7.66335849027280392105033246439, −7.14508611000300556324308840527, −5.97920358744723080760732481721, −5.18719675634285306238155786978, −3.92696382048441280782189319442, −3.32515553684443634556640942630, −2.50465419474535202668698768109, 0.38473371046296143372751368507, 2.12677550590039554762437282010, 3.00504829561743012591582822041, 3.80994254232110065551710715070, 4.88962321417188530006346456083, 6.08321833062587085739504964444, 7.01775091978974021704800066602, 7.49362316433268695130903024126, 8.929930812719282615415489277370, 9.321790078251001894557410668054

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