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Properties

Label 1-3381-3381.188-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.793 - 0.608i$
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  + (0.591 + 0.806i)2-s + (−0.301 + 0.953i)4-s + (0.970 + 0.242i)5-s + (−0.947 + 0.320i)8-s + (0.377 + 0.925i)10-s + (−0.882 + 0.470i)11-s + (−0.488 + 0.872i)13-s + (−0.818 − 0.574i)16-s + (−0.301 − 0.953i)17-s + (0.142 + 0.989i)19-s + (−0.523 + 0.852i)20-s + (−0.900 − 0.433i)22-s + (0.882 + 0.470i)25-s + (−0.992 + 0.122i)26-s + (0.301 + 0.953i)29-s + ⋯
L(s)  = 1  + (0.591 + 0.806i)2-s + (−0.301 + 0.953i)4-s + (0.970 + 0.242i)5-s + (−0.947 + 0.320i)8-s + (0.377 + 0.925i)10-s + (−0.882 + 0.470i)11-s + (−0.488 + 0.872i)13-s + (−0.818 − 0.574i)16-s + (−0.301 − 0.953i)17-s + (0.142 + 0.989i)19-s + (−0.523 + 0.852i)20-s + (−0.900 − 0.433i)22-s + (0.882 + 0.470i)25-s + (−0.992 + 0.122i)26-s + (0.301 + 0.953i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.793 - 0.608i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.793 - 0.608i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4650582240 + 1.371212995i\)
\(L(\frac12)\) \(\approx\) \(-0.4650582240 + 1.371212995i\)
\(L(1)\) \(\approx\) \(0.9363312130 + 0.8898578077i\)
\(L(1)\) \(\approx\) \(0.9363312130 + 0.8898578077i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.591 + 0.806i)T \)
5 \( 1 + (0.970 + 0.242i)T \)
11 \( 1 + (-0.882 + 0.470i)T \)
13 \( 1 + (-0.488 + 0.872i)T \)
17 \( 1 + (-0.301 - 0.953i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (0.301 + 0.953i)T \)
31 \( 1 + (-0.841 - 0.540i)T \)
37 \( 1 + (0.0203 + 0.999i)T \)
41 \( 1 + (0.970 + 0.242i)T \)
43 \( 1 + (0.947 + 0.320i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (-0.986 - 0.162i)T \)
59 \( 1 + (-0.377 - 0.925i)T \)
61 \( 1 + (0.992 + 0.122i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (-0.917 + 0.396i)T \)
73 \( 1 + (0.933 - 0.359i)T \)
79 \( 1 + (-0.959 - 0.281i)T \)
83 \( 1 + (-0.979 - 0.202i)T \)
89 \( 1 + (-0.714 - 0.699i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−18.4030864626316426556093379622, −17.741758777808482637563642307051, −17.35133706043292348008764384671, −16.20325563200124410758727234761, −15.48143073537684721374389864588, −14.812868985661302519411188986481, −14.01348354272911167014975846899, −13.42651648424173571199336557900, −12.79382661489970935941194501136, −12.447393065476585104838738777687, −11.26554382569397471221611296584, −10.65435440328501900023418487804, −10.2073547369773312237022266082, −9.342091462423448801065769239727, −8.753934054752490698829690608987, −7.78365020482462144591851052534, −6.71574255562656297806336236489, −5.78930241727607857418241249764, −5.45913897227110807303671372800, −4.66420949829084530350773858490, −3.76058479850494573200778379675, −2.68529547047643143359138701471, −2.38619948286119158975213114186, −1.31694179305530826513127190962, −0.31383257055690072369512514750, 1.558236768379962942838841467038, 2.51559889676606228105652924430, 3.07009093184099336164994512262, 4.28323935580911226998454454545, 4.90031318536737897536243010695, 5.60439467916943299139722233363, 6.289679400847708299781425125920, 7.10159969653537963508395124120, 7.57356684635538861066939423860, 8.54661859052798124889862888283, 9.43423687829132557411902700786, 9.822890281826342090480485274969, 10.90168642220003990921849720309, 11.67342916286738858775806098080, 12.66626006367346135464548944295, 12.95851702588992753679447713809, 13.955677341831369559646524370735, 14.30222205590477884708332473556, 14.90335572393863658251177443313, 15.94535415883989478417291943263, 16.29609741206867843400664838289, 17.14538775503936912369926181194, 17.74016372313511767642050317400, 18.37048266543748958252342328960, 18.89936188259320052042262191938

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