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Properties

Label 1-392-392.349-r1-0-0
Degree $1$
Conductor $392$
Sign $-0.695 + 0.718i$
Analytic cond. $42.1262$
Root an. cond. $42.1262$
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  + (0.623 + 0.781i)3-s + (0.623 + 0.781i)5-s + (−0.222 + 0.974i)9-s + (0.222 + 0.974i)11-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.900 + 0.433i)17-s + 19-s + (−0.900 + 0.433i)23-s + (−0.222 + 0.974i)25-s + (−0.900 + 0.433i)27-s + (0.900 + 0.433i)29-s − 31-s + (−0.623 + 0.781i)33-s + (0.900 + 0.433i)37-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)3-s + (0.623 + 0.781i)5-s + (−0.222 + 0.974i)9-s + (0.222 + 0.974i)11-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.900 + 0.433i)17-s + 19-s + (−0.900 + 0.433i)23-s + (−0.222 + 0.974i)25-s + (−0.900 + 0.433i)27-s + (0.900 + 0.433i)29-s − 31-s + (−0.623 + 0.781i)33-s + (0.900 + 0.433i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.695 + 0.718i$
Analytic conductor: \(42.1262\)
Root analytic conductor: \(42.1262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 392,\ (1:\ ),\ -0.695 + 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.037838021 + 2.449843970i\)
\(L(\frac12)\) \(\approx\) \(1.037838021 + 2.449843970i\)
\(L(1)\) \(\approx\) \(1.253853467 + 0.7919847211i\)
\(L(1)\) \(\approx\) \(1.253853467 + 0.7919847211i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.900 + 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 - T \)
37 \( 1 + (0.900 + 0.433i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.222 - 0.974i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.222 - 0.974i)T \)
97 \( 1 - T \)
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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

−24.09172605022021849663772232065, −23.38005372657913720874279878162, −21.9413647335827432064010497476, −21.24900500015092036013855535617, −20.31450344934353581138255905195, −19.63130456113751994567212791470, −18.58572410134228959542947142984, −18.00371214593315567626742564212, −16.72328848729862131636337941866, −16.26574149813947896243318032832, −14.73182478121225274052500222401, −13.83851623291224987743187050775, −13.504206581547246925047159556190, −12.197670395992664979084774250643, −11.72760965101189361283402289157, −10.0460005002145232068975368863, −9.17785381380687203016205079272, −8.45068283390982958714587449267, −7.435494038875605858026455175795, −6.30458514817031948219847188954, −5.43680999109131456042928201920, −4.019977907233922909963512347771, −2.78984028360224868962869289022, −1.63941177391818881008385307927, −0.663140022238859562744066171118, 1.65608081966732046349338961444, 2.83148094595167517401120496309, 3.630209632335219785747935933635, 4.97266763971092445296638972248, 5.87654968740033961569013888864, 7.271793275572202262435907501709, 8.04925305161418243010899021658, 9.44912296353768656015116366068, 9.992392874132045136581481454429, 10.692927673104076013769355465044, 11.975268801148460901857611416352, 13.15436516534819071582776923771, 14.155877337843099912051650406911, 14.762858279722591080832974181697, 15.51052978780987257315533160803, 16.56649084074759269106379583977, 17.63234966862630403473529457378, 18.31921272467632487910668636649, 19.55255264634424918853805004619, 20.228743140668675469145466861720, 21.08390911396510185252027255144, 22.06716778597016225814801142251, 22.45207244972547993337661208356, 23.55546432227855323042988488384, 25.002405980632521855552095542644

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