↑

Properties

Label 1-3381-3381.1751-r1-0-0
Degree $1$
Conductor $3381$
Sign $0.747 + 0.664i$
Analytic cond. $363.338$
Root an. cond. $363.338$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.742 + 0.670i)2-s + (0.101 − 0.994i)4-s + (−0.996 − 0.0815i)5-s + (0.591 + 0.806i)8-s + (0.794 − 0.607i)10-s + (−0.986 + 0.162i)11-s + (−0.768 − 0.639i)13-s + (−0.979 − 0.202i)16-s + (−0.101 − 0.994i)17-s + (0.841 − 0.540i)19-s + (−0.182 + 0.983i)20-s + (0.623 − 0.781i)22-s + (0.986 + 0.162i)25-s + (0.999 − 0.0407i)26-s + (−0.101 − 0.994i)29-s + ⋯
L(s)  = 1  + (−0.742 + 0.670i)2-s + (0.101 − 0.994i)4-s + (−0.996 − 0.0815i)5-s + (0.591 + 0.806i)8-s + (0.794 − 0.607i)10-s + (−0.986 + 0.162i)11-s + (−0.768 − 0.639i)13-s + (−0.979 − 0.202i)16-s + (−0.101 − 0.994i)17-s + (0.841 − 0.540i)19-s + (−0.182 + 0.983i)20-s + (0.623 − 0.781i)22-s + (0.986 + 0.162i)25-s + (0.999 − 0.0407i)26-s + (−0.101 − 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.747 + 0.664i$
Analytic conductor: \(363.338\)
Root analytic conductor: \(363.338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (1:\ ),\ 0.747 + 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2088380064 + 0.07941469263i\)
\(L(\frac12)\) \(\approx\) \(0.2088380064 + 0.07941469263i\)
\(L(1)\) \(\approx\) \(0.4541020341 + 0.05886030707i\)
\(L(1)\) \(\approx\) \(0.4541020341 + 0.05886030707i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.742 + 0.670i)T \)
5 \( 1 + (-0.996 - 0.0815i)T \)
11 \( 1 + (-0.986 + 0.162i)T \)
13 \( 1 + (-0.768 - 0.639i)T \)
17 \( 1 + (-0.101 - 0.994i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (-0.101 - 0.994i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + (-0.862 + 0.505i)T \)
41 \( 1 + (-0.996 - 0.0815i)T \)
43 \( 1 + (-0.591 + 0.806i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (0.452 + 0.891i)T \)
59 \( 1 + (-0.794 + 0.607i)T \)
61 \( 1 + (-0.999 - 0.0407i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (0.377 - 0.925i)T \)
73 \( 1 + (-0.992 + 0.122i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (-0.557 + 0.830i)T \)
89 \( 1 + (-0.262 - 0.965i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.761039121155793765495208496287, −18.10739336229641877707902078706, −17.196910944152160649255959032655, −16.56428753562427733397199568537, −15.983358531522699157934686472934, −15.27414276435328004303421235693, −14.48254265093146198226736407774, −13.505336251226825182454325476734, −12.72740239373362288852816601932, −12.117719750036384758979757584723, −11.60953589008632000193508348601, −10.66153287370227838369391338325, −10.39272210482686035656854049730, −9.35406858952196724747543900359, −8.68036456888168790648545434126, −7.926694431480465594005063947430, −7.424987701969614639056570063114, −6.74821399476865391227559464435, −5.47668728051697411080137654414, −4.60473437924337729247336580702, −3.72858571900398907807787585326, −3.194758143936482579478401671200, −2.213002397647519695978660033066, −1.40660002497236398558872952740, −0.15611669370317056412444809262, 0.265094712162573711199504847552, 1.28085210259713062040399783415, 2.54643606960762526007664277998, 3.1846175638818555087190246127, 4.56811842593076969557389300845, 5.03551681149198697699772438223, 5.75029720718775668907961499591, 7.00837925421563582994558566208, 7.34521250574006174582219640305, 7.95506574329329213869515051490, 8.69462525388086571069918137918, 9.47224162260334209880299030754, 10.2333399698978983989662892292, 10.84385291472417362509443351351, 11.697758863761618846114683957170, 12.26514950970230451167877029742, 13.35377195818728409019903015105, 13.94057278424680816293814296260, 14.99407667798009971417574352555, 15.40094405480611120357443934617, 15.88527266062138146743117795268, 16.62080595897038062767420192312, 17.30116324801458456110187742578, 18.241495576817960805504159523433, 18.438936173211711468270909629535

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