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Properties

Label 1-1380-1380.443-r1-0-0
Degree $1$
Conductor $1380$
Sign $-0.999 - 0.0426i$
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  + (0.281 − 0.959i)7-s + (−0.654 + 0.755i)11-s + (−0.281 − 0.959i)13-s + (0.989 − 0.142i)17-s + (−0.142 + 0.989i)19-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (0.909 − 0.415i)37-s + (−0.415 + 0.909i)41-s + (0.540 − 0.841i)43-s − i·47-s + (−0.841 − 0.540i)49-s + (−0.281 + 0.959i)53-s + (0.959 − 0.281i)59-s + (0.841 − 0.540i)61-s + ⋯
L(s)  = 1  + (0.281 − 0.959i)7-s + (−0.654 + 0.755i)11-s + (−0.281 − 0.959i)13-s + (0.989 − 0.142i)17-s + (−0.142 + 0.989i)19-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (0.909 − 0.415i)37-s + (−0.415 + 0.909i)41-s + (0.540 − 0.841i)43-s − i·47-s + (−0.841 − 0.540i)49-s + (−0.281 + 0.959i)53-s + (0.959 − 0.281i)59-s + (0.841 − 0.540i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.999 - 0.0426i$
Analytic conductor: \(148.301\)
Root analytic conductor: \(148.301\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (1:\ ),\ -0.999 - 0.0426i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01132163054 - 0.5304093025i\)
\(L(\frac12)\) \(\approx\) \(0.01132163054 - 0.5304093025i\)
\(L(1)\) \(\approx\) \(0.9226726783 - 0.1514215882i\)
\(L(1)\) \(\approx\) \(0.9226726783 - 0.1514215882i\)

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 \( 1 + (0.281 - 0.959i)T \)
11 \( 1 + (-0.654 + 0.755i)T \)
13 \( 1 + (-0.281 - 0.959i)T \)
17 \( 1 + (0.989 - 0.142i)T \)
19 \( 1 + (-0.142 + 0.989i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (-0.841 + 0.540i)T \)
37 \( 1 + (0.909 - 0.415i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.540 - 0.841i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.281 + 0.959i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (0.755 - 0.654i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (-0.989 - 0.142i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (-0.909 + 0.415i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (-0.909 - 0.415i)T \)
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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

−21.14102661563402838833637454126, −20.37614133357043786628052212049, −19.20171500818954595738916916522, −18.88588520799352983953751262603, −18.096527495071793495678109916647, −17.27208308856682532504141730243, −16.28696072304060718367796177903, −15.88598500542326326503009393483, −14.72560096103031368118700123836, −14.40661820707820862853049107046, −13.24167120134568895061823548141, −12.64992250690331145407021256339, −11.607633791277094544634765503494, −11.218895016205140452589021369878, −10.13161694998620869492837650757, −9.21793214029140515074005536791, −8.61402328707385483023388181333, −7.739795406872404456919572124826, −6.83342880595454759849259355169, −5.78936570533161030899901930352, −5.2370772951216471943167013795, −4.22845477136936209180725392151, −3.06384317275230323627608457232, −2.346539893319864931556718009050, −1.23974611134976598050510860327, 0.10630518601439137692479774074, 1.15375979764103804441737488378, 2.22535181872584900469008396715, 3.32773266178359222668783579140, 4.15794471829752954520361743541, 5.12398925395738597727903895507, 5.83233107381495915457138024063, 7.06961648099233018793783951857, 7.708253198345860266884827554500, 8.242169449888824212973721009666, 9.65216503736341259996761992251, 10.17423119000830585724876975467, 10.806180311118747244684148915398, 11.86145423421682530479202371531, 12.67500266623402511848247781563, 13.28363090320027216576071030021, 14.290349458887128029277516539429, 14.82192103013657087174038298182, 15.711677128088442523627864592971, 16.60654888856158120543585313729, 17.20879122955618888104864407721, 18.00436403949174416253139748567, 18.660268797112794733351627148734, 19.657856255628868952936139758843, 20.42044146828043189610233822826

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