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Properties

Label 1-104-104.45-r1-0-0
Degree $1$
Conductor $104$
Sign $-0.999 + 0.0386i$
Analytic cond. $11.1763$
Root an. cond. $11.1763$
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  + (0.5 + 0.866i)3-s − i·5-s + (−0.866 − 0.5i)7-s + (−0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s − i·21-s + (0.5 + 0.866i)23-s − 25-s − 27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.866 − 0.5i)33-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s − i·5-s + (−0.866 − 0.5i)7-s + (−0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s − i·21-s + (0.5 + 0.866i)23-s − 25-s − 27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.866 − 0.5i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $-0.999 + 0.0386i$
Analytic conductor: \(11.1763\)
Root analytic conductor: \(11.1763\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 104,\ (1:\ ),\ -0.999 + 0.0386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01856619149 + 0.9615167475i\)
\(L(\frac12)\) \(\approx\) \(0.01856619149 + 0.9615167475i\)
\(L(1)\) \(\approx\) \(0.7791197564 + 0.5038713153i\)
\(L(1)\) \(\approx\) \(0.7791197564 + 0.5038713153i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 - iT \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + iT \)
53 \( 1 - T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + T \)
83 \( 1 - iT \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−28.893235135279136294963444254330, −28.38505334767881651528332732640, −26.78072388387718649242246469628, −25.61886188945199126078813679260, −24.96589469544259868752579550237, −23.89269737821990242769792511024, −23.12065185372176799694318017423, −21.46473324496791689374620116308, −20.578817675048993578450454594593, −19.34073722120516712554118467501, −18.79344315007894241742918867801, −17.37128885358502873616147758341, −16.27412345293212775829823189059, −15.11476659067948727405038563158, −13.65937306770214469249603370868, −12.75540443426274746047666034845, −12.16237166704201487090496576473, −10.25941512504828976271250982593, −8.78938358540149870700394433866, −8.19509989172774010019689309431, −6.57379040462667566767028529226, −5.43938193884212680833582736801, −3.53038191371194864398591723760, −2.07126954383236786744123005812, −0.36785347872879074769724840036, 2.613300710192517894419852067235, 3.52865722331341337910665181241, 5.02518686330200669968244721699, 6.69982896006202338419170819752, 7.846345911065288456471503359782, 9.503343595332689398761762661256, 10.24258445154991181840702515909, 11.23136234264449978724732589636, 13.05837320798410591555357894299, 14.07839180734847635945028534902, 15.19857230369041566048695901011, 15.95822015000871238828332547920, 17.22897876246934697269237145282, 18.65171966404113938413692231774, 19.58490874041101567317680340420, 20.65603330539025234159141701851, 21.71883427244563179715421538965, 22.691913956223188525108261458814, 23.470551600731477384864115498453, 25.47299359130401631047554471968, 25.86121118489044684664990423402, 26.797150869079932226345574636431, 27.72943885591666633698688290815, 29.0572198989982306158755374061, 30.01061140166958768438932955460

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