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Properties

Label 2-110-5.4-c1-0-4
Degree $2$
Conductor $110$
Sign $-0.447 + 0.894i$
Analytic cond. $0.878354$
Root an. cond. $0.937205$
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 − i·3-s − 4-s + (−2 − i)5-s − 6-s − 3i·7-s + i·8-s + 2·9-s + (−1 + 2i)10-s + 11-s + i·12-s + 4i·13-s − 3·14-s + (−1 + 2i)15-s + 16-s − 3i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.894 − 0.447i)5-s − 0.408·6-s − 1.13i·7-s + 0.353i·8-s + 0.666·9-s + (−0.316 + 0.632i)10-s + 0.301·11-s + 0.288i·12-s + 1.10i·13-s − 0.801·14-s + (−0.258 + 0.516i)15-s + 0.250·16-s − 0.727i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.475490 - 0.769359i\)
\(L(\frac12)\) \(\approx\) \(0.475490 - 0.769359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (2 + i)T \)
11 \( 1 - T \)
good3 \( 1 + iT - 3T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 - 7iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−13.45559146327761817173452832323, −11.99190237894301110032142643339, −11.64126003200319204649037279643, −10.23030438801077257222125559895, −9.153928025466994275930952749671, −7.69900240295976557573366582433, −6.96274046055113740750404610805, −4.73295666042138592083941452174, −3.67809253555074976153314280703, −1.23979701549011126191742279765, 3.34762356356885543961032986435, 4.76722448442396233580508980739, 6.09299157010974102633359664300, 7.48095748384322237013966681741, 8.465813242576305933897210105602, 9.651846457079291519356964694508, 10.75261710960811921867710425375, 12.03473824417203603620982899101, 12.90607822286981111208006585851, 14.45421972373257204925312642434

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