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Properties

Label 1-1380-1380.239-r0-0-0
Degree $1$
Conductor $1380$
Sign $0.992 + 0.119i$
Analytic cond. $6.40869$
Root an. cond. $6.40869$
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  + (−0.654 − 0.755i)7-s + (0.841 + 0.540i)11-s + (0.654 − 0.755i)13-s + (0.415 + 0.909i)17-s + (−0.415 + 0.909i)19-s + (−0.415 − 0.909i)29-s + (0.142 + 0.989i)31-s + (0.959 + 0.281i)37-s + (0.959 − 0.281i)41-s + (−0.142 + 0.989i)43-s − 47-s + (−0.142 + 0.989i)49-s + (−0.654 − 0.755i)53-s + (−0.654 + 0.755i)59-s + (−0.142 − 0.989i)61-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)7-s + (0.841 + 0.540i)11-s + (0.654 − 0.755i)13-s + (0.415 + 0.909i)17-s + (−0.415 + 0.909i)19-s + (−0.415 − 0.909i)29-s + (0.142 + 0.989i)31-s + (0.959 + 0.281i)37-s + (0.959 − 0.281i)41-s + (−0.142 + 0.989i)43-s − 47-s + (−0.142 + 0.989i)49-s + (−0.654 − 0.755i)53-s + (−0.654 + 0.755i)59-s + (−0.142 − 0.989i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.992 + 0.119i$
Analytic conductor: \(6.40869\)
Root analytic conductor: \(6.40869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (0:\ ),\ 0.992 + 0.119i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.516823315 + 0.09091242814i\)
\(L(\frac12)\) \(\approx\) \(1.516823315 + 0.09091242814i\)
\(L(1)\) \(\approx\) \(1.092920961 + 0.008590344530i\)
\(L(1)\) \(\approx\) \(1.092920961 + 0.008590344530i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (-0.654 - 0.755i)T \)
11 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (0.415 + 0.909i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (0.959 + 0.281i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.654 - 0.755i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (-0.415 + 0.909i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
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

−20.87257429015008229358915242112, −19.97672718953768687934678801166, −19.21203190152152581541931229834, −18.65893827460227343711446300323, −17.944048373512884772831097766727, −16.81868171745308911695762695184, −16.3361406181186682901413171830, −15.57949621171893372122028505866, −14.71954896416991200424606705929, −13.93196188580526799005542489319, −13.17877249773737345359602512846, −12.35046757790912437297687857439, −11.49099620394556997586192565819, −11.02547549472787114249319469788, −9.6535470013925589779748746729, −9.197344934541214044579545664102, −8.52356643750058327005587040463, −7.34696340067048543473158756134, −6.469732426704916465034259857242, −5.91646381371819412211706285061, −4.84845254695399574653196845510, −3.83933382662026814996628178655, −3.00525252959989945672960808115, −2.047296328377048947860028486272, −0.78118551663433027034227802638, 0.92494132877046892520174120334, 1.86879339994429367016255437007, 3.2873317884277377062132786333, 3.82307854648046241758529273057, 4.74042213617219829871251717432, 6.11605045009647052971115685952, 6.377821100201644651396297609777, 7.60012041921333062548868647887, 8.18447720045061219918653833654, 9.30837344118791393704069452329, 10.02781260568251741494142658941, 10.66804860147295842675735577730, 11.583378695633402617086686406257, 12.67075137462085194407534131833, 12.96026618640870061493813117142, 14.0671299977981060425141239412, 14.68426297963800615998492233928, 15.54723849828396658784873276394, 16.41050831559407350096200688977, 17.03978553089504626007384796885, 17.70588445077907268564443772252, 18.64893030494680296579306032110, 19.513780807140669204874771309103, 19.95072674082225939879853664482, 20.82603019502078623492941768021

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