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Properties

Label 1-1104-1104.221-r0-0-0
Degree $1$
Conductor $1104$
Sign $-0.332 - 0.943i$
Analytic cond. $5.12695$
Root an. cond. $5.12695$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.281 + 0.959i)5-s + (−0.654 − 0.755i)7-s + (0.540 − 0.841i)11-s + (−0.755 − 0.654i)13-s + (0.415 + 0.909i)17-s + (−0.909 − 0.415i)19-s + (−0.841 + 0.540i)25-s + (0.909 − 0.415i)29-s + (−0.142 − 0.989i)31-s + (0.540 − 0.841i)35-s + (0.281 − 0.959i)37-s + (−0.959 + 0.281i)41-s + (−0.989 − 0.142i)43-s − 47-s + (−0.142 + 0.989i)49-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)5-s + (−0.654 − 0.755i)7-s + (0.540 − 0.841i)11-s + (−0.755 − 0.654i)13-s + (0.415 + 0.909i)17-s + (−0.909 − 0.415i)19-s + (−0.841 + 0.540i)25-s + (0.909 − 0.415i)29-s + (−0.142 − 0.989i)31-s + (0.540 − 0.841i)35-s + (0.281 − 0.959i)37-s + (−0.959 + 0.281i)41-s + (−0.989 − 0.142i)43-s − 47-s + (−0.142 + 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $-0.332 - 0.943i$
Analytic conductor: \(5.12695\)
Root analytic conductor: \(5.12695\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1104} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1104,\ (0:\ ),\ -0.332 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4533763200 - 0.6404070202i\)
\(L(\frac12)\) \(\approx\) \(0.4533763200 - 0.6404070202i\)
\(L(1)\) \(\approx\) \(0.8729571490 - 0.1049954429i\)
\(L(1)\) \(\approx\) \(0.8729571490 - 0.1049954429i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (0.281 + 0.959i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
11 \( 1 + (0.540 - 0.841i)T \)
13 \( 1 + (-0.755 - 0.654i)T \)
17 \( 1 + (0.415 + 0.909i)T \)
19 \( 1 + (-0.909 - 0.415i)T \)
29 \( 1 + (0.909 - 0.415i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + (0.281 - 0.959i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.755 + 0.654i)T \)
59 \( 1 + (-0.755 - 0.654i)T \)
61 \( 1 + (-0.989 + 0.142i)T \)
67 \( 1 + (-0.540 - 0.841i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (-0.415 + 0.909i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (0.281 - 0.959i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.959 - 0.281i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.59140144737419402283285657172, −20.9345620851115120506530285346, −19.9765164371695537395169141336, −19.48302760543255093024990165142, −18.544230495956237065468301391255, −17.72377673019215488772936747209, −16.805123430837892014756017281592, −16.38626524201968081015065033550, −15.43269607012531505751652929938, −14.631367458884074077580074000952, −13.77170197323304268007207525774, −12.80343778942387815089409693489, −12.17618172413702641436542344153, −11.76494142425876100536407890630, −10.20553409371572510915879964358, −9.610333770634593277823941619018, −8.96064609978872073654255249343, −8.14980173425705734166707910688, −6.90695155857493813115980930481, −6.30171397572166549847560199724, −5.06527711254032738760695706514, −4.66614197365479953063304256712, −3.37515383723905159175947025876, −2.261977916467524274139185377703, −1.40288747537638803278331318319, 0.31208034167677451561107773202, 1.81097681718914046825177455981, 2.99322426571064185769135221823, 3.56461183533906952509113277358, 4.635306340133996242197098318704, 6.05377241905685386697741415632, 6.377197333539680502219054537061, 7.39048974489258887170686623106, 8.17498312081037398768759688592, 9.34518857187129896590878820253, 10.176962543217971530061502656495, 10.675578114329472638577109382010, 11.54065561783922690398787711509, 12.62999674100636194934735186356, 13.36959802198985749367940815473, 14.137976753769284441596649183163, 14.865621159964320788459701441180, 15.57538413683506156605730031062, 16.825323827585316272134397940456, 17.08504979001431112337597756469, 18.06057559221097158288406222948, 19.073228405621223033830215290021, 19.45189263476420387398071926076, 20.21131062987302985550734957753, 21.52708535570935755881162945952

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