Properties

Label 1-967-967.22-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.714 - 0.699i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.807 − 0.589i)2-s + (−0.724 − 0.689i)3-s + (0.304 + 0.952i)4-s + (−0.347 + 0.937i)5-s + (0.177 + 0.984i)6-s + (−0.982 − 0.187i)7-s + (0.316 − 0.948i)8-s + (0.0487 + 0.998i)9-s + (0.833 − 0.552i)10-s + (0.811 + 0.584i)11-s + (0.436 − 0.899i)12-s + (−0.911 − 0.410i)13-s + (0.682 + 0.730i)14-s + (0.898 − 0.439i)15-s + (−0.815 + 0.579i)16-s + (−0.822 − 0.568i)17-s + ⋯
L(s)  = 1  + (−0.807 − 0.589i)2-s + (−0.724 − 0.689i)3-s + (0.304 + 0.952i)4-s + (−0.347 + 0.937i)5-s + (0.177 + 0.984i)6-s + (−0.982 − 0.187i)7-s + (0.316 − 0.948i)8-s + (0.0487 + 0.998i)9-s + (0.833 − 0.552i)10-s + (0.811 + 0.584i)11-s + (0.436 − 0.899i)12-s + (−0.911 − 0.410i)13-s + (0.682 + 0.730i)14-s + (0.898 − 0.439i)15-s + (−0.815 + 0.579i)16-s + (−0.822 − 0.568i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.714 - 0.699i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.714 - 0.699i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06161922786 - 0.1509577690i\)
\(L(\frac12)\) \(\approx\) \(0.06161922786 - 0.1509577690i\)
\(L(1)\) \(\approx\) \(0.3923261582 - 0.08341672142i\)
\(L(1)\) \(\approx\) \(0.3923261582 - 0.08341672142i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.807 - 0.589i)T \)
3 \( 1 + (-0.724 - 0.689i)T \)
5 \( 1 + (-0.347 + 0.937i)T \)
7 \( 1 + (-0.982 - 0.187i)T \)
11 \( 1 + (0.811 + 0.584i)T \)
13 \( 1 + (-0.911 - 0.410i)T \)
17 \( 1 + (-0.822 - 0.568i)T \)
19 \( 1 + (-0.741 + 0.670i)T \)
23 \( 1 + (0.00975 + 0.999i)T \)
29 \( 1 + (0.389 + 0.921i)T \)
31 \( 1 + (-0.767 - 0.641i)T \)
37 \( 1 + (0.643 + 0.765i)T \)
41 \( 1 + (0.460 + 0.887i)T \)
43 \( 1 + (-0.953 + 0.300i)T \)
47 \( 1 + (-0.618 + 0.785i)T \)
53 \( 1 + (-0.158 - 0.987i)T \)
59 \( 1 + (0.571 - 0.820i)T \)
61 \( 1 + (0.538 - 0.842i)T \)
67 \( 1 + (0.938 + 0.344i)T \)
71 \( 1 + (-0.107 - 0.994i)T \)
73 \( 1 + (-0.158 + 0.987i)T \)
79 \( 1 + (-0.927 - 0.374i)T \)
83 \( 1 + (-0.658 + 0.752i)T \)
89 \( 1 + (-0.171 - 0.985i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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

−22.03822417060271130632759697990, −21.39099911288778082438914565115, −20.1558166241378352250270188970, −19.63779558250609125645134117140, −19.00403118166773039546433207930, −17.8088944636797338902456270111, −16.98116303873867732041103753336, −16.67894808729317760179497782187, −15.95464998855233481807286093722, −15.28134531686791636753187025772, −14.48223409255229385532497940776, −13.17007406488153041189091318863, −12.25281047652976895891075645271, −11.49874730679859449207981814132, −10.595246329569373330891109623258, −9.74240173601306417759227005764, −8.95280844487292370182251783762, −8.627746029785839057035111054151, −7.0871255396324825942968000592, −6.422515679607513981642093349364, −5.65571013564025690923823940785, −4.64211905084303563932466026257, −3.92182099459907997897832857208, −2.300193868871150008501537263428, −0.773979778355047853341757380442, 0.1466848157908717079389243498, 1.63082571339910874796090799130, 2.57836671474816831633766209230, 3.49097598805857766910691738122, 4.59089661780416708072263691186, 6.17410211040702803732944254118, 6.871320810116504849947511894037, 7.326078489719015463050612015957, 8.28593305079359121614344405712, 9.70716889960180580618709344055, 10.005577644996224642241820679395, 11.1935946116413358516790848250, 11.54291417669167169786001525710, 12.60847873792972168753024008781, 12.98024942120576058932566791749, 14.218522578414916854165513376485, 15.28950126422826580883510991668, 16.23643740621994883295608651848, 16.9585768874195944503830002566, 17.69722884496385793300386779434, 18.31321172083688095000959957290, 19.157954322147973188872881824967, 19.64842418271241534683261437650, 20.20126598639639457010841600958, 21.82006789996166339045207022061

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