Properties

Label 1-967-967.567-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

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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: $\chi_{967} (567, \cdot )$
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 \( 1 - T \)
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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

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

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