↑

Properties

Label 1-3381-3381.1277-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.938 - 0.345i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 − 0.384i)2-s + (0.704 − 0.709i)4-s + (0.810 + 0.585i)5-s + (0.377 − 0.925i)8-s + (0.973 + 0.229i)10-s + (−0.665 − 0.746i)11-s + (−0.996 − 0.0815i)13-s + (−0.00679 − 0.999i)16-s + (−0.966 + 0.255i)17-s + (−0.981 + 0.189i)19-s + (0.986 − 0.162i)20-s + (−0.900 − 0.433i)22-s + (0.314 + 0.949i)25-s + (−0.951 + 0.307i)26-s + (−0.262 + 0.965i)29-s + ⋯
L(s)  = 1  + (0.923 − 0.384i)2-s + (0.704 − 0.709i)4-s + (0.810 + 0.585i)5-s + (0.377 − 0.925i)8-s + (0.973 + 0.229i)10-s + (−0.665 − 0.746i)11-s + (−0.996 − 0.0815i)13-s + (−0.00679 − 0.999i)16-s + (−0.966 + 0.255i)17-s + (−0.981 + 0.189i)19-s + (0.986 − 0.162i)20-s + (−0.900 − 0.433i)22-s + (0.314 + 0.949i)25-s + (−0.951 + 0.307i)26-s + (−0.262 + 0.965i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.938 - 0.345i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.938 - 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2791673285 - 1.565482181i\)
\(L(\frac12)\) \(\approx\) \(0.2791673285 - 1.565482181i\)
\(L(1)\) \(\approx\) \(1.451160553 - 0.5586255359i\)
\(L(1)\) \(\approx\) \(1.451160553 - 0.5586255359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.923 - 0.384i)T \)
5 \( 1 + (0.810 + 0.585i)T \)
11 \( 1 + (-0.665 - 0.746i)T \)
13 \( 1 + (-0.996 - 0.0815i)T \)
17 \( 1 + (-0.966 + 0.255i)T \)
19 \( 1 + (-0.981 + 0.189i)T \)
29 \( 1 + (-0.262 + 0.965i)T \)
31 \( 1 + (-0.723 - 0.690i)T \)
37 \( 1 + (0.601 - 0.798i)T \)
41 \( 1 + (0.101 - 0.994i)T \)
43 \( 1 + (-0.377 - 0.925i)T \)
47 \( 1 + (-0.733 - 0.680i)T \)
53 \( 1 + (0.997 + 0.0679i)T \)
59 \( 1 + (-0.973 - 0.229i)T \)
61 \( 1 + (-0.209 - 0.977i)T \)
67 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (0.768 - 0.639i)T \)
73 \( 1 + (-0.994 - 0.108i)T \)
79 \( 1 + (0.928 + 0.371i)T \)
83 \( 1 + (0.339 - 0.940i)T \)
89 \( 1 + (-0.833 - 0.552i)T \)
97 \( 1 + (-0.841 + 0.540i)T \)
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

−19.39428252617166738440264819886, −17.93567890235817561935376110360, −17.81910817303050465428518827849, −16.76054744648910102907794065077, −16.54556789972216491496008490884, −15.43123198272601924955107529548, −15.01097551104486240085255288478, −14.29349264938764638748256546731, −13.37740275408996329058621178403, −13.04475482610514542998537296365, −12.43157171078172322510356577720, −11.655587700868054274094836202923, −10.790158863921556143044167517289, −9.98310221676030988441308437976, −9.25618155955831710697687857812, −8.358020374861964122090573604391, −7.63487529262230686775972115564, −6.79370323993380156278472247428, −6.19451902799933284670281966767, −5.32322543061275750488434863440, −4.63656917384904384422014981960, −4.31191035237114763287645392158, −2.8307002672388973779411011961, −2.360637071493779678003585276554, −1.55813589309923754605071667711, 0.2671703911468332342036219285, 1.86123613170133073793191704757, 2.23232850553596868236624733072, 3.07086503823166633107853696752, 3.86101288670493502733536407179, 4.82535076996455585844331220046, 5.52566504984009268647338157916, 6.120833402992838300020718442186, 6.898048374315088567433436410558, 7.547086645085753093660310508728, 8.74878305058977471068840763997, 9.51457461745145715943794307210, 10.45528286752294287860074018630, 10.72165168731203221668146975643, 11.44317552020773987956626415087, 12.47126187818815094219179017438, 12.99491008524464650719020057475, 13.61126770761817390392361050520, 14.28320061949594145105496530028, 14.95479139950655640043725001068, 15.402454725064737795994696519, 16.46959061904453363044162550923, 16.99777672569718180407438800300, 17.99328582760100886576596707103, 18.584901719321209050157746680995

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