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Properties

Label 2-110-11.2-c2-0-3
Degree $2$
Conductor $110$
Sign $-0.582 + 0.812i$
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 + (−4.29 + 3.12i)3-s + (1.61 + 1.17i)4-s + (0.690 + 2.12i)5-s + (7.14 − 2.32i)6-s + (−2.06 + 2.83i)7-s + (−1.66 − 2.28i)8-s + (5.94 − 18.2i)9-s − 3.16i·10-s + (−6.71 − 8.71i)11-s − 10.6·12-s + (−16.7 − 5.45i)13-s + (4.01 − 2.91i)14-s + (−9.61 − 6.98i)15-s + (1.23 + 3.80i)16-s + (26.2 − 8.51i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (−1.43 + 1.04i)3-s + (0.404 + 0.293i)4-s + (0.138 + 0.425i)5-s + (1.19 − 0.387i)6-s + (−0.294 + 0.405i)7-s + (−0.207 − 0.286i)8-s + (0.660 − 2.03i)9-s − 0.316i·10-s + (−0.610 − 0.791i)11-s − 0.885·12-s + (−1.29 − 0.419i)13-s + (0.286 − 0.208i)14-s + (−0.640 − 0.465i)15-s + (0.0772 + 0.237i)16-s + (1.54 − 0.500i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $-0.582 + 0.812i$
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: \((2,\ 110,\ (\ :1),\ -0.582 + 0.812i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0256898 - 0.0500048i\)
\(L(\frac12)\) \(\approx\) \(0.0256898 - 0.0500048i\)
\(L(2)\) 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 \( 1 + (6.71 + 8.71i)T \)
good3 \( 1 + (4.29 - 3.12i)T + (2.78 - 8.55i)T^{2} \)
7 \( 1 + (2.06 - 2.83i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (16.7 + 5.45i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-26.2 + 8.51i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (5.64 + 7.77i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + 17.4T + 529T^{2} \)
29 \( 1 + (-7.46 + 10.2i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (13.7 - 42.1i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (56.3 + 40.9i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (7.27 + 10.0i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 5.67iT - 1.84e3T^{2} \)
47 \( 1 + (15.5 - 11.3i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (1.11 - 3.44i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (49.6 + 36.0i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (84.0 - 27.3i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 71.7T + 4.48e3T^{2} \)
71 \( 1 + (7.40 + 22.7i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (37.0 - 51.0i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (43.4 + 14.1i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-123. + 40.0i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 3.64T + 7.92e3T^{2} \)
97 \( 1 + (-49.9 + 153. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−12.43619609690579747837281554676, −11.85948667722858752237025446782, −10.61447774246775211798047656695, −10.22171310748464900218426049240, −9.167532632138710507054464239892, −7.45567410451338784148213009029, −6.02018052402277706687085013994, −5.10751199028121401972875026961, −3.18415832130567462417692123645, −0.05851460972206457985396268785, 1.72191079804069974840197738184, 4.96063743913427027291414097469, 6.04434963420265650667102764243, 7.20229459855235201979336192560, 7.905113212013623282716929231955, 9.845248206691786827049288538412, 10.48762838151572823020494938099, 12.02564562225886217056711484384, 12.31667983551721265639282353160, 13.48881962435896778267246696511

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