Properties

Label 2-110-55.7-c1-0-0
Degree $2$
Conductor $110$
Sign $0.0886 - 0.996i$
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  + (−0.987 + 0.156i)2-s + (0.992 + 1.94i)3-s + (0.951 − 0.309i)4-s + (0.244 + 2.22i)5-s + (−1.28 − 1.76i)6-s + (−3.05 − 1.55i)7-s + (−0.891 + 0.453i)8-s + (−1.04 + 1.43i)9-s + (−0.589 − 2.15i)10-s + (3.24 + 0.694i)11-s + (1.54 + 1.54i)12-s + (0.387 + 2.44i)13-s + (3.26 + 1.06i)14-s + (−4.08 + 2.68i)15-s + (0.809 − 0.587i)16-s + (−0.196 + 1.24i)17-s + ⋯
L(s)  = 1  + (−0.698 + 0.110i)2-s + (0.572 + 1.12i)3-s + (0.475 − 0.154i)4-s + (0.109 + 0.994i)5-s + (−0.524 − 0.721i)6-s + (−1.15 − 0.588i)7-s + (−0.315 + 0.160i)8-s + (−0.347 + 0.478i)9-s + (−0.186 − 0.682i)10-s + (0.977 + 0.209i)11-s + (0.446 + 0.446i)12-s + (0.107 + 0.679i)13-s + (0.872 + 0.283i)14-s + (−1.05 + 0.692i)15-s + (0.202 − 0.146i)16-s + (−0.0477 + 0.301i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0886 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0886 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.0886 - 0.996i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ 0.0886 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.642297 + 0.587682i\)
\(L(\frac12)\) \(\approx\) \(0.642297 + 0.587682i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.987 - 0.156i)T \)
5 \( 1 + (-0.244 - 2.22i)T \)
11 \( 1 + (-3.24 - 0.694i)T \)
good3 \( 1 + (-0.992 - 1.94i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (3.05 + 1.55i)T + (4.11 + 5.66i)T^{2} \)
13 \( 1 + (-0.387 - 2.44i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (0.196 - 1.24i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (-1.62 + 5.00i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-5.42 + 5.42i)T - 23iT^{2} \)
29 \( 1 + (-0.472 - 1.45i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.34 + 4.61i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.60 + 5.11i)T + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (-3.98 - 1.29i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-0.832 - 0.832i)T + 43iT^{2} \)
47 \( 1 + (4.97 - 2.53i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (9.00 - 1.42i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (-2.65 + 0.862i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.34 - 8.73i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (4.03 + 4.03i)T + 67iT^{2} \)
71 \( 1 + (7.25 - 5.27i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.14 + 8.12i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (11.2 + 8.19i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.9 + 1.89i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 - 4.89iT - 89T^{2} \)
97 \( 1 + (-0.442 - 2.79i)T + (-92.2 + 29.9i)T^{2} \)
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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

−14.37619121399570561554099359222, −13.00589256882360767528595845365, −11.31607777373193951665456246438, −10.49839990327176361086795421828, −9.509272920363315537842372836588, −9.066854972325833087731270017867, −7.17933378924105675955591061824, −6.42847480788112744267540732725, −4.14140845708374506621238262687, −2.96701524217947346251993523377, 1.40732525469904195243511262032, 3.22947545964520374859585120255, 5.73488695716220337464837260059, 6.94994301522794341668895670237, 8.115129283331720781366293968765, 9.029876465736283916127371204737, 9.800438618276861323951034413061, 11.60094244458260487703236581464, 12.61906257844318263271024603321, 13.03649737905573623508886899768

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