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Properties

Label 1-3381-3381.260-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.215 - 0.976i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (0.301 − 0.953i)2-s + (−0.818 − 0.574i)4-s + (0.882 + 0.470i)5-s + (−0.794 + 0.607i)8-s + (0.714 − 0.699i)10-s + (0.557 − 0.830i)11-s + (−0.523 − 0.852i)13-s + (0.339 + 0.940i)16-s + (−0.818 + 0.574i)17-s + (0.959 − 0.281i)19-s + (−0.452 − 0.891i)20-s + (−0.623 − 0.781i)22-s + (0.557 + 0.830i)25-s + (−0.970 + 0.242i)26-s + (0.818 − 0.574i)29-s + ⋯
L(s)  = 1  + (0.301 − 0.953i)2-s + (−0.818 − 0.574i)4-s + (0.882 + 0.470i)5-s + (−0.794 + 0.607i)8-s + (0.714 − 0.699i)10-s + (0.557 − 0.830i)11-s + (−0.523 − 0.852i)13-s + (0.339 + 0.940i)16-s + (−0.818 + 0.574i)17-s + (0.959 − 0.281i)19-s + (−0.452 − 0.891i)20-s + (−0.623 − 0.781i)22-s + (0.557 + 0.830i)25-s + (−0.970 + 0.242i)26-s + (0.818 − 0.574i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.215 - 0.976i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (260, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.215 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.365107077 - 1.699453662i\)
\(L(\frac12)\) \(\approx\) \(1.365107077 - 1.699453662i\)
\(L(1)\) \(\approx\) \(1.153260569 - 0.6988737472i\)
\(L(1)\) \(\approx\) \(1.153260569 - 0.6988737472i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.301 - 0.953i)T \)
5 \( 1 + (0.882 + 0.470i)T \)
11 \( 1 + (0.557 - 0.830i)T \)
13 \( 1 + (-0.523 - 0.852i)T \)
17 \( 1 + (-0.818 + 0.574i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (0.818 - 0.574i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (0.999 - 0.0407i)T \)
41 \( 1 + (-0.882 - 0.470i)T \)
43 \( 1 + (-0.794 - 0.607i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (0.947 + 0.320i)T \)
59 \( 1 + (0.714 - 0.699i)T \)
61 \( 1 + (-0.970 - 0.242i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (-0.685 + 0.728i)T \)
73 \( 1 + (0.742 - 0.670i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (0.917 + 0.396i)T \)
89 \( 1 + (0.0203 + 0.999i)T \)
97 \( 1 + (0.654 - 0.755i)T \)
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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

−18.67534294749581811106672436503, −18.07961294425675143829274922539, −17.564661278216103572799811610298, −16.74832731147495773606440586782, −16.483310383697289148223034906846, −15.544791705117007506401818757910, −14.81025292090044217959937586920, −14.19613250447158493424777243067, −13.54880293644616837759135867676, −13.057233497143236412896794410785, −12.02233158893596870243561822431, −11.7641716401874091972158143150, −10.278393530740654854536155097724, −9.531888378362716026313070673325, −9.2238869014578222769004346480, −8.38278520882188981923840731208, −7.44318017259560947678562853717, −6.76450085868068375545365025786, −6.23638550896401201281326460952, −5.26933532173867370587514468308, −4.69468299816261854164604934416, −4.11787758086602380084594297417, −2.89836183896328281945389675086, −2.02014799026412043552023522657, −0.95095238586809200276695115161, 0.70085907447739161143451588051, 1.56359578012489873302379708258, 2.54039319009098686958035829858, 3.03825521519985263188941671113, 3.88474823131879301832526414712, 4.87734189051770012340756269748, 5.57774924215786664984680347126, 6.22105423774025997520866234051, 7.03024669658806961496392405304, 8.24989247327440114064237970786, 8.899505279194535988256728039784, 9.666528158949992844225916946387, 10.30562559566299780162298759262, 10.83612784280561994957261074957, 11.62925579574758982585067680732, 12.27988841229636489123064641472, 13.19516034060886014246354475268, 13.65499671664069806640395050190, 14.21662725234467451090683790219, 14.9840705777004767390021784302, 15.63847848389172625579710795625, 16.8129457915351584902021313869, 17.54588450845724731256843333256, 17.91371740912808704323342557298, 18.73166800305391136378970343043

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