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Properties

Label 1-967-967.436-r1-0-0
Degree $1$
Conductor $967$
Sign $0.934 - 0.356i$
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  + (0.436 − 0.899i)2-s + (−0.874 + 0.485i)3-s + (−0.618 − 0.785i)4-s + (−0.0617 − 0.998i)5-s + (0.0552 + 0.998i)6-s + (−0.840 − 0.541i)7-s + (−0.977 + 0.212i)8-s + (0.527 − 0.849i)9-s + (−0.924 − 0.380i)10-s + (−0.107 + 0.994i)11-s + (0.922 + 0.386i)12-s + (−0.914 − 0.404i)13-s + (−0.854 + 0.519i)14-s + (0.538 + 0.842i)15-s + (−0.235 + 0.971i)16-s + (−0.799 − 0.600i)17-s + ⋯
L(s)  = 1  + (0.436 − 0.899i)2-s + (−0.874 + 0.485i)3-s + (−0.618 − 0.785i)4-s + (−0.0617 − 0.998i)5-s + (0.0552 + 0.998i)6-s + (−0.840 − 0.541i)7-s + (−0.977 + 0.212i)8-s + (0.527 − 0.849i)9-s + (−0.924 − 0.380i)10-s + (−0.107 + 0.994i)11-s + (0.922 + 0.386i)12-s + (−0.914 − 0.404i)13-s + (−0.854 + 0.519i)14-s + (0.538 + 0.842i)15-s + (−0.235 + 0.971i)16-s + (−0.799 − 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.934 - 0.356i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (436, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.934 - 0.356i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04288681920 + 0.007915731550i\)
\(L(\frac12)\) \(\approx\) \(0.04288681920 + 0.007915731550i\)
\(L(1)\) \(\approx\) \(0.4283672150 - 0.3949970814i\)
\(L(1)\) \(\approx\) \(0.4283672150 - 0.3949970814i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.436 - 0.899i)T \)
3 \( 1 + (-0.874 + 0.485i)T \)
5 \( 1 + (-0.0617 - 0.998i)T \)
7 \( 1 + (-0.840 - 0.541i)T \)
11 \( 1 + (-0.107 + 0.994i)T \)
13 \( 1 + (-0.914 - 0.404i)T \)
17 \( 1 + (-0.799 - 0.600i)T \)
19 \( 1 + (0.310 - 0.950i)T \)
23 \( 1 + (-0.494 - 0.869i)T \)
29 \( 1 + (-0.710 + 0.703i)T \)
31 \( 1 + (-0.687 + 0.726i)T \)
37 \( 1 + (0.597 - 0.801i)T \)
41 \( 1 + (-0.203 - 0.979i)T \)
43 \( 1 + (-0.880 + 0.474i)T \)
47 \( 1 + (-0.701 + 0.712i)T \)
53 \( 1 + (-0.829 + 0.557i)T \)
59 \( 1 + (0.728 - 0.684i)T \)
61 \( 1 + (-0.949 - 0.313i)T \)
67 \( 1 + (-0.972 + 0.232i)T \)
71 \( 1 + (0.560 - 0.828i)T \)
73 \( 1 + (-0.829 - 0.557i)T \)
79 \( 1 + (0.0812 + 0.996i)T \)
83 \( 1 + (-0.383 + 0.923i)T \)
89 \( 1 + (0.285 - 0.958i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−21.926487179437946782412343508976, −21.49636586128311245751405061478, −19.72593684030063154185253503486, −18.800993510705734451604485096950, −18.51832372844956370516867190236, −17.55824689017750218110536904801, −16.733913596028282768612088204342, −16.168715697091344048002743080736, −15.27181341934530285726089843597, −14.588193842607338293424638831918, −13.43694448380918128846980120388, −13.128211308876091074191885845892, −11.84715779032925176292632629415, −11.57412775051746171392710589456, −10.24700218124393561301821338929, −9.45871117086757392896050293896, −8.14859150254208967670513378860, −7.4400437111123866173985100302, −6.509887997102559483272577997974, −6.07853179750181403138414400697, −5.36526947247819937345805146365, −4.06329547160435346959326316180, −3.16585412602215339072628726226, −2.04749338837320149697738370599, −0.020723198294455288031220463848, 0.40850629956636674536052215749, 1.701430152124326272133520143075, 2.96953485985608921920896031851, 4.11936719967519854251503065418, 4.74428045642247695435697000362, 5.302245867145803187304227690243, 6.46235605909297676750393024348, 7.37942907259408836565042488652, 9.06947624187640814625896137514, 9.50595064155686291935467733930, 10.262708735740029569301223059235, 11.061147518740159489850806831553, 12.00833390573692735029161440526, 12.72619107497702525353427068092, 12.99522538227383949089879166973, 14.23359498388597474116090810870, 15.28623065289157641884432071858, 15.95763625971751577821981812512, 16.8078253677031473724922202565, 17.65445575460645418541299967746, 18.210419530252281272345981880288, 19.60252474084662431816922391750, 20.11661686520588388327047728924, 20.568658624384533851899633851609, 21.61312984600938595219012935992

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