↑

Properties

Label 1-967-967.249-r1-0-0
Degree $1$
Conductor $967$
Sign $0.972 - 0.232i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (0.983 + 0.181i)2-s + (0.990 − 0.136i)3-s + (0.934 + 0.356i)4-s + (0.949 − 0.313i)5-s + (0.998 + 0.0455i)6-s + (−0.803 − 0.595i)7-s + (0.854 + 0.519i)8-s + (0.962 − 0.269i)9-s + (0.990 − 0.136i)10-s + (0.962 − 0.269i)11-s + (0.974 + 0.225i)12-s + (0.247 + 0.968i)13-s + (−0.682 − 0.730i)14-s + (0.898 − 0.439i)15-s + (0.746 + 0.665i)16-s + (−0.0682 − 0.997i)17-s + ⋯
L(s)  = 1  + (0.983 + 0.181i)2-s + (0.990 − 0.136i)3-s + (0.934 + 0.356i)4-s + (0.949 − 0.313i)5-s + (0.998 + 0.0455i)6-s + (−0.803 − 0.595i)7-s + (0.854 + 0.519i)8-s + (0.962 − 0.269i)9-s + (0.990 − 0.136i)10-s + (0.962 − 0.269i)11-s + (0.974 + 0.225i)12-s + (0.247 + 0.968i)13-s + (−0.682 − 0.730i)14-s + (0.898 − 0.439i)15-s + (0.746 + 0.665i)16-s + (−0.0682 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.972 - 0.232i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.972 - 0.232i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(8.250128271 - 0.9719790836i\)
\(L(\frac12)\) \(\approx\) \(8.250128271 - 0.9719790836i\)
\(L(1)\) \(\approx\) \(3.342611905 - 0.1196256230i\)
\(L(1)\) \(\approx\) \(3.342611905 - 0.1196256230i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.983 + 0.181i)T \)
3 \( 1 + (0.990 - 0.136i)T \)
5 \( 1 + (0.949 - 0.313i)T \)
7 \( 1 + (-0.803 - 0.595i)T \)
11 \( 1 + (0.962 - 0.269i)T \)
13 \( 1 + (0.247 + 0.968i)T \)
17 \( 1 + (-0.0682 - 0.997i)T \)
19 \( 1 + (-0.377 + 0.926i)T \)
23 \( 1 + (0.775 - 0.631i)T \)
29 \( 1 + (-0.962 - 0.269i)T \)
31 \( 1 + (0.0227 - 0.999i)T \)
37 \( 1 + (0.247 + 0.968i)T \)
41 \( 1 + (-0.460 - 0.887i)T \)
43 \( 1 + (0.829 + 0.557i)T \)
47 \( 1 + (-0.898 + 0.439i)T \)
53 \( 1 + (-0.158 - 0.987i)T \)
59 \( 1 + (-0.158 + 0.987i)T \)
61 \( 1 + (0.538 - 0.842i)T \)
67 \( 1 + (-0.854 + 0.519i)T \)
71 \( 1 + (-0.334 + 0.942i)T \)
73 \( 1 + (-0.158 + 0.987i)T \)
79 \( 1 + (0.158 - 0.987i)T \)
83 \( 1 + (-0.998 - 0.0455i)T \)
89 \( 1 + (0.877 + 0.480i)T \)
97 \( 1 + T \)
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

−21.61161620918054111805827569678, −21.04283444662386489700726996497, −19.933376401110276471817548201486, −19.60000698293632368087822093815, −18.74043963236839129119303621285, −17.65911598128917496841569945874, −16.67416521580983637788103055008, −15.62251307106896297439189050024, −15.00072920995940904956988405074, −14.5254094353949216520951216809, −13.46483333132054491167878633860, −13.026321399702077056234053875040, −12.39609699876911100545962357986, −11.02512920828201257270282778837, −10.30428789243394748995966828408, −9.42444484139960895139377028555, −8.80078443301556917564811920548, −7.38497312182070788982447277759, −6.590949882464696443978844795444, −5.85443625734878362856845592321, −4.87135170898149384511284985351, −3.63659848687059123562565921668, −3.094605023342991602667840922540, −2.18928923105627007600298398531, −1.35078152455216379400834268386, 1.13769606212938297036329733607, 2.049186397286360242587374365878, 2.99961126552467666180439283750, 3.90898431115614115133206208394, 4.58141301259951567843424252267, 5.9568029584386903737761053249, 6.62469307201197910559944519807, 7.2756517994914133918341005777, 8.507702712039397675248528722899, 9.37412966465672269768975084098, 10.013504288855504465529074236, 11.21302458793301330570392999466, 12.23806417095472327317331698931, 13.14595513348040467925766625901, 13.51133845496068814332041721151, 14.31540352785413662918800873803, 14.73958244637682760472908589249, 16.05226966037553883493940263217, 16.569348316375875939669972162870, 17.255868172840889113141094678282, 18.736148143338047699427180891769, 19.23799746232925723826578563163, 20.44517261703270887713295389133, 20.59447182786386421542694447439, 21.46560446287827289866458622806

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