↑

Properties

Label 1-967-967.13-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.999 + 0.0283i$
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  + (0.663 − 0.748i)2-s + (−0.653 − 0.756i)3-s + (−0.120 − 0.992i)4-s + (0.857 − 0.514i)5-s + (−0.999 − 0.0130i)6-s + (0.0422 + 0.999i)7-s + (−0.822 − 0.568i)8-s + (−0.145 + 0.989i)9-s + (0.184 − 0.982i)10-s + (−0.297 − 0.954i)11-s + (−0.672 + 0.739i)12-s + (0.677 + 0.735i)13-s + (0.775 + 0.631i)14-s + (−0.949 − 0.313i)15-s + (−0.971 + 0.238i)16-s + (0.241 + 0.970i)17-s + ⋯
L(s)  = 1  + (0.663 − 0.748i)2-s + (−0.653 − 0.756i)3-s + (−0.120 − 0.992i)4-s + (0.857 − 0.514i)5-s + (−0.999 − 0.0130i)6-s + (0.0422 + 0.999i)7-s + (−0.822 − 0.568i)8-s + (−0.145 + 0.989i)9-s + (0.184 − 0.982i)10-s + (−0.297 − 0.954i)11-s + (−0.672 + 0.739i)12-s + (0.677 + 0.735i)13-s + (0.775 + 0.631i)14-s + (−0.949 − 0.313i)15-s + (−0.971 + 0.238i)16-s + (0.241 + 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.999 + 0.0283i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.999 + 0.0283i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03584228613 - 2.527683213i\)
\(L(\frac12)\) \(\approx\) \(0.03584228613 - 2.527683213i\)
\(L(1)\) \(\approx\) \(0.9664477908 - 1.002058644i\)
\(L(1)\) \(\approx\) \(0.9664477908 - 1.002058644i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.663 - 0.748i)T \)
3 \( 1 + (-0.653 - 0.756i)T \)
5 \( 1 + (0.857 - 0.514i)T \)
7 \( 1 + (0.0422 + 0.999i)T \)
11 \( 1 + (-0.297 - 0.954i)T \)
13 \( 1 + (0.677 + 0.735i)T \)
17 \( 1 + (0.241 + 0.970i)T \)
19 \( 1 + (0.993 + 0.110i)T \)
23 \( 1 + (0.0292 - 0.999i)T \)
29 \( 1 + (0.932 - 0.362i)T \)
31 \( 1 + (-0.999 - 0.00650i)T \)
37 \( 1 + (-0.867 + 0.497i)T \)
41 \( 1 + (0.990 - 0.136i)T \)
43 \( 1 + (0.383 - 0.923i)T \)
47 \( 1 + (0.0941 - 0.995i)T \)
53 \( 1 + (-0.998 + 0.0455i)T \)
59 \( 1 + (0.266 - 0.963i)T \)
61 \( 1 + (0.377 - 0.926i)T \)
67 \( 1 + (-0.494 + 0.869i)T \)
71 \( 1 + (0.316 - 0.948i)T \)
73 \( 1 + (-0.998 - 0.0455i)T \)
79 \( 1 + (0.587 + 0.809i)T \)
83 \( 1 + (-0.895 - 0.445i)T \)
89 \( 1 + (-0.505 - 0.862i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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

−22.26474185582863881604241859414, −21.14554232392617570425439355221, −20.81357035939191898485770357046, −19.98658501394196780020284640918, −18.11951394750980775502872311670, −17.81741446078594849487664473312, −17.2619289253434132821123819037, −16.14015941535063536484954904715, −15.83425093783406805977539995858, −14.74089083777217372496463401044, −14.13422973924201627203426356450, −13.34505619489599771386224091936, −12.52813451252682134532318812793, −11.43228264514580073297169137749, −10.68886716490370249512313141379, −9.81792861585169239144344225368, −9.14757281522005302338223007921, −7.584184912007270964974170072715, −7.10527530430100991407333677294, −6.08069479394053675042043560388, −5.3501276535982298694111996656, −4.66729140709532563983782219524, −3.59073255770221602827709027293, −2.86516625833845710828724344285, −1.11134902663523407178857655981, 0.49389815200527143601710438837, 1.5212466018714054585137062238, 2.167280571252837660950591719804, 3.23708585522016301910755150785, 4.62260849599807874785259650183, 5.56373761470123043463747296662, 5.907772078846834425175199529250, 6.69001792133269970795205160454, 8.35032003925333794800984165960, 8.93686401696491464453701896152, 10.07673631162561167308081400681, 10.90836206702714500525292047104, 11.67653005922737014813758266861, 12.44829928598462478341542433517, 12.963366406423736769042261005779, 13.87546386895604677568291172218, 14.29838035730965939102361289173, 15.734776887666661440756432025676, 16.37103679011733531406359479565, 17.41143246158631168102055886998, 18.362796326143799914179954002655, 18.70061310789835118068964314740, 19.47989993730616367868355238567, 20.63606780071726925057275487569, 21.30136917444059914130826963023

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