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Properties

Label 1-3381-3381.779-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.986 - 0.163i$
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  + (0.833 − 0.552i)2-s + (0.390 − 0.920i)4-s + (−0.00679 + 0.999i)5-s + (−0.182 − 0.983i)8-s + (0.546 + 0.837i)10-s + (0.511 + 0.859i)11-s + (−0.979 + 0.202i)13-s + (−0.694 − 0.719i)16-s + (0.601 − 0.798i)17-s + (0.888 + 0.458i)19-s + (0.917 + 0.396i)20-s + (0.900 + 0.433i)22-s + (−0.999 − 0.0135i)25-s + (−0.704 + 0.709i)26-s + (0.992 + 0.122i)29-s + ⋯
L(s)  = 1  + (0.833 − 0.552i)2-s + (0.390 − 0.920i)4-s + (−0.00679 + 0.999i)5-s + (−0.182 − 0.983i)8-s + (0.546 + 0.837i)10-s + (0.511 + 0.859i)11-s + (−0.979 + 0.202i)13-s + (−0.694 − 0.719i)16-s + (0.601 − 0.798i)17-s + (0.888 + 0.458i)19-s + (0.917 + 0.396i)20-s + (0.900 + 0.433i)22-s + (−0.999 − 0.0135i)25-s + (−0.704 + 0.709i)26-s + (0.992 + 0.122i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.986 - 0.163i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.986 - 0.163i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.922930164 - 0.2403151528i\)
\(L(\frac12)\) \(\approx\) \(2.922930164 - 0.2403151528i\)
\(L(1)\) \(\approx\) \(1.716000459 - 0.2811879729i\)
\(L(1)\) \(\approx\) \(1.716000459 - 0.2811879729i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.833 - 0.552i)T \)
5 \( 1 + (-0.00679 + 0.999i)T \)
11 \( 1 + (0.511 + 0.859i)T \)
13 \( 1 + (-0.979 + 0.202i)T \)
17 \( 1 + (0.601 - 0.798i)T \)
19 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (0.992 + 0.122i)T \)
31 \( 1 + (-0.327 - 0.945i)T \)
37 \( 1 + (0.675 - 0.737i)T \)
41 \( 1 + (0.862 + 0.505i)T \)
43 \( 1 + (-0.182 + 0.983i)T \)
47 \( 1 + (-0.955 + 0.294i)T \)
53 \( 1 + (-0.169 - 0.985i)T \)
59 \( 1 + (0.546 + 0.837i)T \)
61 \( 1 + (0.966 - 0.255i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (-0.986 - 0.162i)T \)
73 \( 1 + (0.963 - 0.268i)T \)
79 \( 1 + (-0.580 + 0.814i)T \)
83 \( 1 + (0.996 - 0.0815i)T \)
89 \( 1 + (0.994 + 0.108i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−18.98481420789700125978977920691, −17.73379889267030597587613028054, −17.316031307694930309926750828479, −16.58234981775721735986460295624, −16.14244094703615411960541549357, −15.41634503890329900210971019774, −14.589818136844951584799902088604, −14.00202686437420235942477467896, −13.34094055720076329032364168700, −12.59336094966081784175150058687, −12.06089621656655256541434697948, −11.50305218357408329063320184399, −10.47059687568734178610955411212, −9.51096786366942178632280703122, −8.72948791963500926635867054429, −8.12970853554655511789632056786, −7.42608596785970758395994477116, −6.53780634354126567754888564716, −5.75069176095585659669057211226, −5.14163903978926798878415709725, −4.50773461455720003902852650706, −3.61099530305172792047005874210, −2.9417424180476186971797572301, −1.82847499018185752110051125462, −0.7641797490255919334127185914, 0.94520490333932326594317217887, 2.03967018508853568643506349586, 2.66046794075208581971784843694, 3.3871620508180014627410872721, 4.238724696743471865548486957920, 4.948332565050568084363478873127, 5.79263037447657177030684399096, 6.58830350676745493563775576962, 7.24443308862651015583799446851, 7.81645458851731826615926157968, 9.46855806078780810328195383839, 9.71834192241256251793929400704, 10.36845611779780561986471280725, 11.44020810087350770828828479160, 11.690556872844824736674466291795, 12.46414710938127599464735725955, 13.22349640578193982347141322077, 14.16090665177382343387743723526, 14.565157253501843662063583654638, 14.922687441112110780991419413427, 15.95324770443710790797206790579, 16.50082719829944480227989547548, 17.765450152246169416624731459014, 18.091797714802610590374953675283, 19.04454892193911522751203090283

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