Properties

Label 1-967-967.140-r0-0-0
Degree $1$
Conductor $967$
Sign $0.481 - 0.876i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (0.892 − 0.451i)2-s + (0.316 + 0.948i)3-s + (0.592 − 0.805i)4-s + (0.560 − 0.828i)5-s + (0.710 + 0.703i)6-s + (−0.822 − 0.568i)7-s + (0.165 − 0.986i)8-s + (−0.799 + 0.600i)9-s + (0.126 − 0.991i)10-s + (0.763 + 0.646i)11-s + (0.951 + 0.307i)12-s + (0.763 − 0.646i)13-s + (−0.990 − 0.136i)14-s + (0.962 + 0.269i)15-s + (−0.297 − 0.954i)16-s + (−0.477 − 0.878i)17-s + ⋯
L(s)  = 1  + (0.892 − 0.451i)2-s + (0.316 + 0.948i)3-s + (0.592 − 0.805i)4-s + (0.560 − 0.828i)5-s + (0.710 + 0.703i)6-s + (−0.822 − 0.568i)7-s + (0.165 − 0.986i)8-s + (−0.799 + 0.600i)9-s + (0.126 − 0.991i)10-s + (0.763 + 0.646i)11-s + (0.951 + 0.307i)12-s + (0.763 − 0.646i)13-s + (−0.990 − 0.136i)14-s + (0.962 + 0.269i)15-s + (−0.297 − 0.954i)16-s + (−0.477 − 0.878i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.481 - 0.876i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.481 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.594729359 - 1.534947513i\)
\(L(\frac12)\) \(\approx\) \(2.594729359 - 1.534947513i\)
\(L(1)\) \(\approx\) \(1.945284002 - 0.5624537200i\)
\(L(1)\) \(\approx\) \(1.945284002 - 0.5624537200i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.892 - 0.451i)T \)
3 \( 1 + (0.316 + 0.948i)T \)
5 \( 1 + (0.560 - 0.828i)T \)
7 \( 1 + (-0.822 - 0.568i)T \)
11 \( 1 + (0.763 + 0.646i)T \)
13 \( 1 + (0.763 - 0.646i)T \)
17 \( 1 + (-0.477 - 0.878i)T \)
19 \( 1 + (0.938 + 0.344i)T \)
23 \( 1 + (-0.184 + 0.982i)T \)
29 \( 1 + (-0.967 + 0.250i)T \)
31 \( 1 + (0.924 + 0.380i)T \)
37 \( 1 + (0.981 + 0.193i)T \)
41 \( 1 + (-0.334 - 0.942i)T \)
43 \( 1 + (0.0487 - 0.998i)T \)
47 \( 1 + (0.811 - 0.584i)T \)
53 \( 1 + (-0.917 - 0.398i)T \)
59 \( 1 + (-0.883 - 0.468i)T \)
61 \( 1 + (-0.334 - 0.942i)T \)
67 \( 1 + (0.924 - 0.380i)T \)
71 \( 1 + (0.892 + 0.451i)T \)
73 \( 1 + (-0.917 + 0.398i)T \)
79 \( 1 + (0.999 + 0.0390i)T \)
83 \( 1 + (-0.844 + 0.536i)T \)
89 \( 1 + (0.924 - 0.380i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−22.093083905454414377167277005699, −21.37620511125686791933084845945, −20.34234938322448230015818592087, −19.453813232655839904065682777468, −18.73497429219793849646554763815, −18.037290059391267368130151309440, −17.072027127806606434166163784891, −16.32506526320703440195649020363, −15.26713770783929457749058387190, −14.60233303811639711900671137163, −13.79506580605789635261932212899, −13.352586426373791403397715719412, −12.53557223623805959350942099347, −11.59298706023536767257175374039, −11.029651530456471982359477972006, −9.49532984334133497261205320365, −8.73185269002416873721294321726, −7.75282963964451775657430573738, −6.70463182661398306834729220695, −6.21154679925384761446009254913, −5.87071739915850747586843319340, −4.16160829807926525680985844923, −3.18451086587103703613415308734, −2.59790002265281118637756006196, −1.53531939598389598258889045194, 0.95826154708571495477029352468, 2.15297158278311792199402591308, 3.35346091543955622782601730117, 3.884783600686664396062489312956, 4.869819436480702180298024564123, 5.55551528945846698741908901438, 6.46954484563591542848168164656, 7.62489796373215393599888877945, 9.06529915736493962120898726064, 9.60961298636125419369418089330, 10.19204888558748618243499005038, 11.1816654030736966978551779283, 12.063512345860785355384170041160, 12.98895480145192289541151107213, 13.76549130526884557568959305977, 14.111143429781332959255891085962, 15.484090795858017367241402946874, 15.780114552929091376442635756954, 16.66844780241190013655051247169, 17.43871946685689954288887768852, 18.71469039449688615244317395047, 19.95576435341302941451054818712, 20.19927887013669734653389190149, 20.65816726529544208391470816002, 21.6594637726308283495445322957

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