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Properties

Label 2-110-11.2-c2-0-4
Degree $2$
Conductor $110$
Sign $0.999 - 0.0443i$
Analytic cond. $2.99728$
Root an. cond. $1.73126$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 + 0.437i)2-s + (2.27 − 1.65i)3-s + (1.61 + 1.17i)4-s + (0.690 + 2.12i)5-s + (3.77 − 1.22i)6-s + (0.859 − 1.18i)7-s + (1.66 + 2.28i)8-s + (−0.342 + 1.05i)9-s + 3.16i·10-s + (2.69 − 10.6i)11-s + 5.61·12-s + (−7.50 − 2.43i)13-s + (1.67 − 1.21i)14-s + (5.08 + 3.69i)15-s + (1.23 + 3.80i)16-s + (−7.18 + 2.33i)17-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (0.757 − 0.550i)3-s + (0.404 + 0.293i)4-s + (0.138 + 0.425i)5-s + (0.629 − 0.204i)6-s + (0.122 − 0.169i)7-s + (0.207 + 0.286i)8-s + (−0.0380 + 0.117i)9-s + 0.316i·10-s + (0.245 − 0.969i)11-s + 0.468·12-s + (−0.577 − 0.187i)13-s + (0.119 − 0.0868i)14-s + (0.338 + 0.246i)15-s + (0.0772 + 0.237i)16-s + (−0.422 + 0.137i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0443i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.999 - 0.0443i$
Analytic conductor: \(2.99728\)
Root analytic conductor: \(1.73126\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1),\ 0.999 - 0.0443i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.37809 + 0.0527427i\)
\(L(\frac12)\) \(\approx\) \(2.37809 + 0.0527427i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 - 0.437i)T \)
5 \( 1 + (-0.690 - 2.12i)T \)
11 \( 1 + (-2.69 + 10.6i)T \)
good3 \( 1 + (-2.27 + 1.65i)T + (2.78 - 8.55i)T^{2} \)
7 \( 1 + (-0.859 + 1.18i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (7.50 + 2.43i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (7.18 - 2.33i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (0.318 + 0.438i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + 21.5T + 529T^{2} \)
29 \( 1 + (14.6 - 20.1i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-1.82 + 5.61i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (2.72 + 1.97i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (21.0 + 28.9i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 65.2iT - 1.84e3T^{2} \)
47 \( 1 + (24.9 - 18.0i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (-17.3 + 53.4i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-89.8 - 65.2i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (7.44 - 2.41i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 36.7T + 4.48e3T^{2} \)
71 \( 1 + (-33.1 - 101. i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-81.3 + 111. i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-89.3 - 29.0i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-2.54 + 0.827i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 3.30T + 7.92e3T^{2} \)
97 \( 1 + (-47.6 + 146. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−13.69402737154699155545429103098, −12.70052391666475810963415684006, −11.49239577047524431904413917578, −10.42893221726093872843872943272, −8.814208772711417902612542755911, −7.79568795379536698418268943745, −6.77186491718086024074695098072, −5.42221877811142502681915401066, −3.64458737952430193212390784841, −2.26508436933791598313713932392, 2.25306274577149045738527945129, 3.89069199844259438094449113825, 4.94503168333931127287389194828, 6.52486329157254556193742137988, 8.046246541953881865151204575913, 9.363946116396252933729383717586, 10.00960096929986033089640325280, 11.58555555938283190320686643185, 12.41909371129375440827881453724, 13.50655549529460540397595087694

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