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Properties

Label 2-110-11.3-c1-0-0
Degree $2$
Conductor $110$
Sign $-0.220 - 0.975i$
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  + (−0.809 + 0.587i)2-s + (1 + 3.07i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−2.61 − 1.90i)6-s + (0.809 − 2.48i)7-s + (0.309 + 0.951i)8-s + (−6.04 + 4.39i)9-s − 10-s + (−2.54 − 2.12i)11-s + 3.23·12-s + (1.30 − 0.951i)13-s + (0.809 + 2.48i)14-s + (−1 + 3.07i)15-s + (−0.809 − 0.587i)16-s + (4.23 + 3.07i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.577 + 1.77i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (−1.06 − 0.776i)6-s + (0.305 − 0.941i)7-s + (0.109 + 0.336i)8-s + (−2.01 + 1.46i)9-s − 0.316·10-s + (−0.767 − 0.641i)11-s + 0.934·12-s + (0.363 − 0.263i)13-s + (0.216 + 0.665i)14-s + (−0.258 + 0.794i)15-s + (−0.202 − 0.146i)16-s + (1.02 + 0.746i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $-0.220 - 0.975i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ -0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.603531 + 0.755440i\)
\(L(\frac12)\) \(\approx\) \(0.603531 + 0.755440i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (2.54 + 2.12i)T \)
good3 \( 1 + (-1 - 3.07i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.809 + 2.48i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.30 + 0.951i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.23 - 3.07i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.26 + 3.88i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.145T + 23T^{2} \)
29 \( 1 + (0.381 - 1.17i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.85 + 4.25i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.263 - 0.812i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.572 + 1.76i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.23T + 43T^{2} \)
47 \( 1 + (-3.5 - 10.7i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.736 - 0.534i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.736 + 2.26i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.23 + 5.25i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 0.763T + 67T^{2} \)
71 \( 1 + (10.7 + 7.77i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.527 - 1.62i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.23 + 0.898i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (12.7 + 9.23i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + (-9.70 + 7.05i)T + (29.9 - 92.2i)T^{2} \)
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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

−14.24129415487598630950886074540, −13.46365730799103962294448377100, −11.13612388410584567383626327300, −10.50779665986759223933459870856, −9.850950899887139770989817838847, −8.662980336039716331505079858087, −7.78732002411149714953887795002, −5.87165781259701693233645598767, −4.60448263139425405692379786559, −3.14377541079798043747284821099, 1.63120457434034122612308164467, 2.79218766861194229476159210542, 5.62095413985784400899036655819, 6.98091356769265814758596819409, 8.070063971714528121583775495351, 8.725274363632144712685088363542, 10.01163188693441557956723195924, 11.79382528359755014480301140282, 12.24473314173808180428992358915, 13.18023162403324771957452460501

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