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Properties

Label 1-967-967.215-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.485 + 0.873i$
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  + (−0.995 + 0.0974i)2-s + (−0.260 − 0.965i)3-s + (0.981 − 0.193i)4-s + (−0.998 − 0.0585i)5-s + (0.353 + 0.935i)6-s + (0.710 + 0.703i)7-s + (−0.957 + 0.288i)8-s + (−0.864 + 0.502i)9-s + (0.999 − 0.0390i)10-s + (−0.145 + 0.989i)11-s + (−0.442 − 0.896i)12-s + (−0.145 − 0.989i)13-s + (−0.775 − 0.631i)14-s + (0.203 + 0.979i)15-s + (0.924 − 0.380i)16-s + (−0.638 − 0.769i)17-s + ⋯
L(s)  = 1  + (−0.995 + 0.0974i)2-s + (−0.260 − 0.965i)3-s + (0.981 − 0.193i)4-s + (−0.998 − 0.0585i)5-s + (0.353 + 0.935i)6-s + (0.710 + 0.703i)7-s + (−0.957 + 0.288i)8-s + (−0.864 + 0.502i)9-s + (0.999 − 0.0390i)10-s + (−0.145 + 0.989i)11-s + (−0.442 − 0.896i)12-s + (−0.145 − 0.989i)13-s + (−0.775 − 0.631i)14-s + (0.203 + 0.979i)15-s + (0.924 − 0.380i)16-s + (−0.638 − 0.769i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.485 + 0.873i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.485 + 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07845427834 + 0.1333877310i\)
\(L(\frac12)\) \(\approx\) \(0.07845427834 + 0.1333877310i\)
\(L(1)\) \(\approx\) \(0.4639546406 - 0.06460922090i\)
\(L(1)\) \(\approx\) \(0.4639546406 - 0.06460922090i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.995 + 0.0974i)T \)
3 \( 1 + (-0.260 - 0.965i)T \)
5 \( 1 + (-0.998 - 0.0585i)T \)
7 \( 1 + (0.710 + 0.703i)T \)
11 \( 1 + (-0.145 + 0.989i)T \)
13 \( 1 + (-0.145 - 0.989i)T \)
17 \( 1 + (-0.638 - 0.769i)T \)
19 \( 1 + (0.653 + 0.756i)T \)
23 \( 1 + (-0.407 - 0.913i)T \)
29 \( 1 + (0.996 + 0.0779i)T \)
31 \( 1 + (-0.822 - 0.568i)T \)
37 \( 1 + (-0.297 + 0.954i)T \)
41 \( 1 + (-0.990 + 0.136i)T \)
43 \( 1 + (0.892 + 0.451i)T \)
47 \( 1 + (-0.608 - 0.793i)T \)
53 \( 1 + (0.460 - 0.887i)T \)
59 \( 1 + (-0.0292 - 0.999i)T \)
61 \( 1 + (-0.990 + 0.136i)T \)
67 \( 1 + (-0.822 + 0.568i)T \)
71 \( 1 + (-0.995 - 0.0974i)T \)
73 \( 1 + (0.460 + 0.887i)T \)
79 \( 1 + (-0.967 + 0.250i)T \)
83 \( 1 + (-0.511 + 0.859i)T \)
89 \( 1 + (-0.822 + 0.568i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−21.46721351017748974037227407197, −20.59031483933003059419291874390, −19.7068656134848805568384071740, −19.460187131492088722205046298713, −18.21693597922700060013208076391, −17.48208818242101681180669817793, −16.71393069349216784733091553900, −16.05325532347774013191749543248, −15.51922760488805580143495024432, −14.60301232143501947923910684169, −13.72855787170819423646087221624, −12.13256405925017854135244379140, −11.49679663273629941901731221927, −10.91338102063268665736027374505, −10.40725417677575217312971315456, −9.1194778519399338281482639769, −8.67005158731558584401886169272, −7.706608910747853876659195760431, −6.92590858904554709433701138636, −5.8131180969924655835887987918, −4.59607281334975053181041526988, −3.80853356275913209760038008654, −2.95464444277724493981886873046, −1.42670639819678713384496117775, −0.10830615944025389049887549435, 1.176798252457688938834277063568, 2.21553244134491086230269957828, 3.03029032840810932979868575850, 4.75167133410422962718011153304, 5.59648495350063365473020745104, 6.73841223210080742610634983433, 7.43397563780317715101908365613, 8.14157077362090637844575490599, 8.59830976592295781321762973647, 9.87402667713810970924220713322, 10.83958487348149642595077788843, 11.70071952663950111661653665205, 12.10785836562618305212228049559, 12.85551792399273043904352276125, 14.32003274873743617733341183393, 15.05976182043115487280321969272, 15.757357991643611271831895422462, 16.624554047952496304782559938005, 17.593436399486578233018403696503, 18.24560302744681342932402127914, 18.52238453976830641624677478270, 19.59307503825419453952856870886, 20.257841968293219606391061363442, 20.67558524935715764897131709045, 22.25535020826072420483857061521

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