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Properties

Label 2-110-11.10-c2-0-0
Degree $2$
Conductor $110$
Sign $-0.400 - 0.916i$
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  − 1.41i·2-s − 2.29·3-s − 2.00·4-s − 2.23·5-s + 3.24i·6-s + 4.07i·7-s + 2.82i·8-s − 3.73·9-s + 3.16i·10-s + (−10.0 + 4.40i)11-s + 4.59·12-s + 22.8i·13-s + 5.76·14-s + 5.13·15-s + 4.00·16-s − 5.68i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.765·3-s − 0.500·4-s − 0.447·5-s + 0.541i·6-s + 0.581i·7-s + 0.353i·8-s − 0.414·9-s + 0.316i·10-s + (−0.916 + 0.400i)11-s + 0.382·12-s + 1.75i·13-s + 0.411·14-s + 0.342·15-s + 0.250·16-s − 0.334i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $-0.400 - 0.916i$
Analytic conductor: \(2.99728\)
Root analytic conductor: \(1.73126\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1),\ -0.400 - 0.916i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.125884 + 0.192394i\)
\(L(\frac12)\) \(\approx\) \(0.125884 + 0.192394i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
5 \( 1 + 2.23T \)
11 \( 1 + (10.0 - 4.40i)T \)
good3 \( 1 + 2.29T + 9T^{2} \)
7 \( 1 - 4.07iT - 49T^{2} \)
13 \( 1 - 22.8iT - 169T^{2} \)
17 \( 1 + 5.68iT - 289T^{2} \)
19 \( 1 + 29.9iT - 361T^{2} \)
23 \( 1 + 32.8T + 529T^{2} \)
29 \( 1 + 2.84iT - 841T^{2} \)
31 \( 1 + 46.2T + 961T^{2} \)
37 \( 1 - 17.0T + 1.36e3T^{2} \)
41 \( 1 + 35.5iT - 1.68e3T^{2} \)
43 \( 1 - 45.5iT - 1.84e3T^{2} \)
47 \( 1 + 56.9T + 2.20e3T^{2} \)
53 \( 1 - 62.1T + 2.80e3T^{2} \)
59 \( 1 + 11.4T + 3.48e3T^{2} \)
61 \( 1 - 85.6iT - 3.72e3T^{2} \)
67 \( 1 - 5.61T + 4.48e3T^{2} \)
71 \( 1 - 25.2T + 5.04e3T^{2} \)
73 \( 1 - 125. iT - 5.32e3T^{2} \)
79 \( 1 - 40.4iT - 6.24e3T^{2} \)
83 \( 1 + 8.92iT - 6.88e3T^{2} \)
89 \( 1 + 44.8T + 7.92e3T^{2} \)
97 \( 1 + 141.T + 9.40e3T^{2} \)
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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

−13.54412820725982909712413623388, −12.35872498379345603647757232601, −11.59261949159113058246264380792, −10.99477125048591066818748595722, −9.591601291775433927835315199015, −8.576295622126358048268908064062, −7.02228853286123214081184481106, −5.53444320165869103049078972138, −4.36658654029195635797139938410, −2.42438123531777725463289021039, 0.17581810111552357558762431813, 3.58664649497141681594911789661, 5.31262193897051069889146654542, 6.05080658459074223733295588386, 7.69865630012629200774870433307, 8.261541776238746011642505940042, 10.16280742522723437161419537528, 10.81968478759584836895367126418, 12.18463033588901097213433164415, 13.06972513888412136295104623585

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