Properties

Label 1-3381-3381.221-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.440 + 0.897i$
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  + (−0.390 + 0.920i)2-s + (−0.694 − 0.719i)4-s + (−0.999 − 0.0135i)5-s + (0.933 − 0.359i)8-s + (0.403 − 0.915i)10-s + (−0.476 + 0.879i)11-s + (0.917 − 0.396i)13-s + (−0.0339 + 0.999i)16-s + (−0.275 − 0.961i)17-s + (−0.580 − 0.814i)19-s + (0.685 + 0.728i)20-s + (−0.623 − 0.781i)22-s + (0.999 + 0.0271i)25-s + (0.00679 + 0.999i)26-s + (−0.970 − 0.242i)29-s + ⋯
L(s)  = 1  + (−0.390 + 0.920i)2-s + (−0.694 − 0.719i)4-s + (−0.999 − 0.0135i)5-s + (0.933 − 0.359i)8-s + (0.403 − 0.915i)10-s + (−0.476 + 0.879i)11-s + (0.917 − 0.396i)13-s + (−0.0339 + 0.999i)16-s + (−0.275 − 0.961i)17-s + (−0.580 − 0.814i)19-s + (0.685 + 0.728i)20-s + (−0.623 − 0.781i)22-s + (0.999 + 0.0271i)25-s + (0.00679 + 0.999i)26-s + (−0.970 − 0.242i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.440 + 0.897i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.440 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3108682049 + 0.4990781116i\)
\(L(\frac12)\) \(\approx\) \(0.3108682049 + 0.4990781116i\)
\(L(1)\) \(\approx\) \(0.5711881352 + 0.2370705523i\)
\(L(1)\) \(\approx\) \(0.5711881352 + 0.2370705523i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.390 + 0.920i)T \)
5 \( 1 + (-0.999 - 0.0135i)T \)
11 \( 1 + (-0.476 + 0.879i)T \)
13 \( 1 + (0.917 - 0.396i)T \)
17 \( 1 + (-0.275 - 0.961i)T \)
19 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (-0.970 - 0.242i)T \)
31 \( 1 + (-0.786 + 0.618i)T \)
37 \( 1 + (0.0882 + 0.996i)T \)
41 \( 1 + (-0.488 - 0.872i)T \)
43 \( 1 + (0.933 + 0.359i)T \)
47 \( 1 + (-0.826 + 0.563i)T \)
53 \( 1 + (-0.942 + 0.333i)T \)
59 \( 1 + (0.403 - 0.915i)T \)
61 \( 1 + (-0.869 + 0.494i)T \)
67 \( 1 + (0.888 + 0.458i)T \)
71 \( 1 + (-0.947 - 0.320i)T \)
73 \( 1 + (0.855 - 0.517i)T \)
79 \( 1 + (0.327 + 0.945i)T \)
83 \( 1 + (0.986 - 0.162i)T \)
89 \( 1 + (0.976 + 0.215i)T \)
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

−18.78778538308993291387506529769, −18.24184212346682687829834437387, −17.275818215117892439547774902860, −16.41207182345101618768546174777, −16.19783323691986873843426589482, −15.08335163027335513804000313756, −14.42991685226341185023046296616, −13.44421627472196649432180340126, −12.91034734236143350826653392223, −12.26381377229186598699509337676, −11.365919486126753808064637534654, −10.965819342958317306506272739409, −10.47252309856863571895989824074, −9.34550841826107240153119317881, −8.67325984776152739783499602725, −8.11282465218444551284784877818, −7.55411279449913420977120996105, −6.42856828201199761137484299047, −5.56300098482300157541170348734, −4.49415381451870097444367229984, −3.67340839442302494543831901643, −3.48730707713062019308923708180, −2.2448960622648870159342854318, −1.44065758488181231555476828843, −0.31557041730140984929581698956, 0.6847333563092311624742444887, 1.82587382520285022730501184285, 3.0328416302590410136997738786, 3.98316798336669476808206295393, 4.73283920983681356438675193726, 5.28836439945094688220233974083, 6.37622604321690412747261280666, 7.030018905713279721632973045842, 7.652211615822418619790858846348, 8.26335554197084452222929523320, 9.04455973329777810236696656586, 9.64008999956096836198706992584, 10.76866491629808395234073169758, 11.03440960416324686523535322939, 12.1332122914365130040762468727, 12.970338356937805728131811921203, 13.48151002389939702057524700473, 14.49034563211806990796389853802, 15.0938737725499339914341823585, 15.74209534420858294789295082624, 16.010871659504480017494748484307, 16.94080246557690571241544922704, 17.68543930438397023328030691336, 18.28774133791760314721603200151, 18.85620073212630529999343297563

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