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Properties

Label 1-1380-1380.419-r1-0-0
Degree $1$
Conductor $1380$
Sign $-0.992 + 0.125i$
Analytic cond. $148.301$
Root an. cond. $148.301$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.992 + 0.125i$
Analytic conductor: \(148.301\)
Root analytic conductor: \(148.301\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (1:\ ),\ -0.992 + 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02715725961 + 0.4326101210i\)
\(L(\frac12)\) \(\approx\) \(0.02715725961 + 0.4326101210i\)
\(L(1)\) \(\approx\) \(0.9581490864 + 0.08373579586i\)
\(L(1)\) \(\approx\) \(0.9581490864 + 0.08373579586i\)

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

−20.43888239428717612484433616761, −19.49413570527037836595656796481, −18.63159852782194062333475446284, −17.99619175668570229997984888873, −17.54961645091890086034891983114, −16.294403994381522062444662861036, −15.62065404089779335400450803885, −15.20769660140303765500380400794, −14.01295306585631530109618485584, −13.49629649620105094229178417100, −12.60791847300736764418310043886, −11.69903755484216341799828346045, −11.05556047562603318964245309306, −10.33745705982566786901286733767, −9.16876114921030812001374784827, −8.591080356657075710263642792469, −7.78517960380020631702042822908, −6.92992175864813189219059848507, −5.67515852585736250680733248363, −5.331092717974277237626999458586, −4.28914769841275162329815235970, −3.019602168238677677542534993513, −2.45769446091245898211487933941, −1.170220904873374393391980840503, −0.0828486531372095187004886885, 1.36353108407807758589599017185, 1.98926328820673436101560695988, 3.36783866299625982976749099134, 4.194361515128781234097596024995, 4.951141100529436936325687818524, 5.96971095150495535838891816800, 6.87176371201567347087388572947, 7.77197927036759534515121137263, 8.31569610300733457811764398029, 9.40001356027388793547706859875, 10.27335191312566508656047669074, 10.935483670696478276389599815926, 11.63905751307264036584642746496, 12.71513417027159391242866972078, 13.31501166338413007154718373761, 14.21908604455381986459727375954, 14.79384025277978079727662927848, 15.75411939392225365801518001999, 16.4887761040751628437632560912, 17.238546097366460698629291358931, 18.0296672948069050898553619529, 18.612997641704324387358975725865, 19.56655374359727532327637862326, 20.38348596910942653147692410840, 20.9618545941944926917568125210

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