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Properties

Label 1-1113-1113.2-r0-0-0
Degree $1$
Conductor $1113$
Sign $-0.683 + 0.729i$
Analytic cond. $5.16875$
Root an. cond. $5.16875$
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  + (0.391 + 0.919i)2-s + (−0.692 + 0.721i)4-s + (0.903 + 0.428i)5-s + (−0.935 − 0.354i)8-s + (−0.0402 + 0.999i)10-s + (0.948 − 0.316i)11-s + (−0.970 + 0.239i)13-s + (−0.0402 − 0.999i)16-s + (0.987 + 0.160i)17-s + (−0.721 + 0.692i)19-s + (−0.935 + 0.354i)20-s + (0.663 + 0.748i)22-s + (0.866 − 0.5i)23-s + (0.632 + 0.774i)25-s + (−0.600 − 0.799i)26-s + ⋯
L(s)  = 1  + (0.391 + 0.919i)2-s + (−0.692 + 0.721i)4-s + (0.903 + 0.428i)5-s + (−0.935 − 0.354i)8-s + (−0.0402 + 0.999i)10-s + (0.948 − 0.316i)11-s + (−0.970 + 0.239i)13-s + (−0.0402 − 0.999i)16-s + (0.987 + 0.160i)17-s + (−0.721 + 0.692i)19-s + (−0.935 + 0.354i)20-s + (0.663 + 0.748i)22-s + (0.866 − 0.5i)23-s + (0.632 + 0.774i)25-s + (−0.600 − 0.799i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1113\)    =    \(3 \cdot 7 \cdot 53\)
Sign: $-0.683 + 0.729i$
Analytic conductor: \(5.16875\)
Root analytic conductor: \(5.16875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1113} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1113,\ (0:\ ),\ -0.683 + 0.729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7844027917 + 1.809243361i\)
\(L(\frac12)\) \(\approx\) \(0.7844027917 + 1.809243361i\)
\(L(1)\) \(\approx\) \(1.081208775 + 0.8873182880i\)
\(L(1)\) \(\approx\) \(1.081208775 + 0.8873182880i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
53 \( 1 \)
good2 \( 1 + (0.391 + 0.919i)T \)
5 \( 1 + (0.903 + 0.428i)T \)
11 \( 1 + (0.948 - 0.316i)T \)
13 \( 1 + (-0.970 + 0.239i)T \)
17 \( 1 + (0.987 + 0.160i)T \)
19 \( 1 + (-0.721 + 0.692i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.748 + 0.663i)T \)
31 \( 1 + (0.979 - 0.200i)T \)
37 \( 1 + (0.0402 + 0.999i)T \)
41 \( 1 + (-0.663 + 0.748i)T \)
43 \( 1 + (-0.885 + 0.464i)T \)
47 \( 1 + (0.996 + 0.0804i)T \)
59 \( 1 + (0.428 + 0.903i)T \)
61 \( 1 + (-0.160 - 0.987i)T \)
67 \( 1 + (-0.721 - 0.692i)T \)
71 \( 1 + (0.464 + 0.885i)T \)
73 \( 1 + (-0.160 + 0.987i)T \)
79 \( 1 + (0.600 + 0.799i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.632 - 0.774i)T \)
97 \( 1 + (0.568 + 0.822i)T \)
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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

−21.0245806005902952755929637796, −20.52790963645094971391776564571, −19.508133592654915263852221454722, −19.13132973934129129803053457426, −17.97737485856707892162908281635, −17.26346343947757499679239257359, −16.8120840175392021714316687513, −15.28730615481730737963082301400, −14.680475397159603543475753382050, −13.86598977002534345744270786316, −13.20006457544122418077802443075, −12.34610601552762729203843410414, −11.84352572595878780090810497264, −10.73162220639333632090027540154, −9.94473506132282213771593262250, −9.348227290697381684329338157621, −8.65954635932825562871511944182, −7.27026964794945224314759122464, −6.230945963259226449772658156368, −5.33559146256965413785264716690, −4.70146213173988257316742723935, −3.667461154418635111760742547954, −2.57593130714219084856002199972, −1.80601945811875761662894440588, −0.76284309244941405009693122431, 1.34433964085959165282167987506, 2.686016111663722057809125205808, 3.55753543928829601663599173377, 4.65166382651080449412464779344, 5.4780508368433545071003634305, 6.36848340766195767986309801990, 6.8513028764976921086445763035, 7.885312550036082099123501603867, 8.81725192255816854957163847406, 9.61376522147777327962856166785, 10.3255268714100560274099070591, 11.6122367976266234451975998133, 12.41352803914074560718298817150, 13.22339063846938927307906426107, 14.07546900393177450038574460396, 14.67590329989914156408428250153, 15.07927485768150749833091370355, 16.54239653426670269747206611175, 16.915821494488870786631740014807, 17.42323108293144843953367095922, 18.6551394378719615758845284482, 18.91482746490606139254320995208, 20.27161161192048205560558272342, 21.326752600232519762930009327, 21.68641113768440146961939255816

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