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Properties

Label 1-1392-1392.35-r0-0-0
Degree $1$
Conductor $1392$
Sign $0.912 - 0.409i$
Analytic cond. $6.46442$
Root an. cond. $6.46442$
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  + (0.974 − 0.222i)5-s + (−0.900 − 0.433i)7-s + (0.781 + 0.623i)11-s + (0.781 + 0.623i)13-s + 17-s + (−0.433 − 0.900i)19-s + (0.222 − 0.974i)23-s + (0.900 − 0.433i)25-s + (−0.222 − 0.974i)31-s + (−0.974 − 0.222i)35-s + (−0.781 + 0.623i)37-s − 41-s + (0.974 + 0.222i)43-s + (−0.623 + 0.781i)47-s + (0.623 + 0.781i)49-s + ⋯
L(s)  = 1  + (0.974 − 0.222i)5-s + (−0.900 − 0.433i)7-s + (0.781 + 0.623i)11-s + (0.781 + 0.623i)13-s + 17-s + (−0.433 − 0.900i)19-s + (0.222 − 0.974i)23-s + (0.900 − 0.433i)25-s + (−0.222 − 0.974i)31-s + (−0.974 − 0.222i)35-s + (−0.781 + 0.623i)37-s − 41-s + (0.974 + 0.222i)43-s + (−0.623 + 0.781i)47-s + (0.623 + 0.781i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1392 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1392 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1392\)    =    \(2^{4} \cdot 3 \cdot 29\)
Sign: $0.912 - 0.409i$
Analytic conductor: \(6.46442\)
Root analytic conductor: \(6.46442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1392} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1392,\ (0:\ ),\ 0.912 - 0.409i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.863473156 - 0.3985745224i\)
\(L(\frac12)\) \(\approx\) \(1.863473156 - 0.3985745224i\)
\(L(1)\) \(\approx\) \(1.280214968 - 0.1151206848i\)
\(L(1)\) \(\approx\) \(1.280214968 - 0.1151206848i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
29 \( 1 \)
good5 \( 1 + (0.974 - 0.222i)T \)
7 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (0.781 + 0.623i)T \)
13 \( 1 + (0.781 + 0.623i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.433 - 0.900i)T \)
23 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + (-0.781 + 0.623i)T \)
41 \( 1 - T \)
43 \( 1 + (0.974 + 0.222i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (0.974 - 0.222i)T \)
59 \( 1 - iT \)
61 \( 1 + (-0.433 + 0.900i)T \)
67 \( 1 + (0.781 - 0.623i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (0.623 + 0.781i)T \)
83 \( 1 + (-0.433 - 0.900i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
97 \( 1 + (0.900 - 0.433i)T \)
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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

−21.199012704137678831518111237337, −20.05401144287768433497919402324, −19.23121825977560867384243607666, −18.64171400831118505273076409382, −17.90703194034382613045505371215, −16.99760144096286990667077303179, −16.437590375356277294217797262108, −15.59720233543990145510943790787, −14.67595121606143066141670162220, −13.94913814589384943869991697460, −13.276830714177221554861388627637, −12.49348130597411413814672600964, −11.72612025316262162150343967170, −10.55846309831352595249056845608, −10.11689553352520217138082666050, −9.13061603374080360683990703948, −8.642050000863935267973291868330, −7.41881506169366935241958022714, −6.41914501526076845466896991935, −5.87672776423777441475390978046, −5.26375253279129640902901400755, −3.53731161047463319700895710117, −3.33001701320755723273615632153, −2.00254621391911022381295756624, −1.07268948402725572524647282006, 0.89280906998977617342459231531, 1.85896971837480118931559170568, 2.894884688511227796108018424209, 3.91967247033651495286399196474, 4.73886021126516322077088585215, 5.86678035126820660110593732996, 6.54994754819625293950010973868, 7.10979853188580138965729050175, 8.46796446824525540958853723102, 9.22427998077557543296531866928, 9.82637677234026983493265668868, 10.53726464644150651107811351313, 11.53458491473128177453940929742, 12.52949171411135378926014267392, 13.09128485156474049714514989062, 13.87989527608981695991190262009, 14.49971319048060630873431012192, 15.48267610011929376838271035269, 16.4897938433385591325203120399, 16.89256556813402388633262895794, 17.59975508147824546001687902011, 18.5980667190721566653836609749, 19.16628117914827869973888911520, 20.13929532628007763265339529335, 20.72823395951405733092880533454

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