↑

Properties

Label 1-967-967.453-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.840 + 0.542i$
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  + (−0.992 − 0.123i)2-s + (0.999 + 0.0195i)3-s + (0.969 + 0.244i)4-s + (0.941 + 0.337i)5-s + (−0.989 − 0.142i)6-s + (−0.448 − 0.893i)7-s + (−0.932 − 0.362i)8-s + (0.999 + 0.0390i)9-s + (−0.892 − 0.451i)10-s + (−0.184 − 0.982i)11-s + (0.964 + 0.263i)12-s + (−0.943 + 0.331i)13-s + (0.334 + 0.942i)14-s + (0.934 + 0.356i)15-s + (0.880 + 0.474i)16-s + (−0.442 + 0.896i)17-s + ⋯
L(s)  = 1  + (−0.992 − 0.123i)2-s + (0.999 + 0.0195i)3-s + (0.969 + 0.244i)4-s + (0.941 + 0.337i)5-s + (−0.989 − 0.142i)6-s + (−0.448 − 0.893i)7-s + (−0.932 − 0.362i)8-s + (0.999 + 0.0390i)9-s + (−0.892 − 0.451i)10-s + (−0.184 − 0.982i)11-s + (0.964 + 0.263i)12-s + (−0.943 + 0.331i)13-s + (0.334 + 0.942i)14-s + (0.934 + 0.356i)15-s + (0.880 + 0.474i)16-s + (−0.442 + 0.896i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.840 + 0.542i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (453, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.840 + 0.542i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1337182040 + 0.4535820830i\)
\(L(\frac12)\) \(\approx\) \(0.1337182040 + 0.4535820830i\)
\(L(1)\) \(\approx\) \(0.8902947484 + 0.003482607449i\)
\(L(1)\) \(\approx\) \(0.8902947484 + 0.003482607449i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.992 - 0.123i)T \)
3 \( 1 + (0.999 + 0.0195i)T \)
5 \( 1 + (0.941 + 0.337i)T \)
7 \( 1 + (-0.448 - 0.893i)T \)
11 \( 1 + (-0.184 - 0.982i)T \)
13 \( 1 + (-0.943 + 0.331i)T \)
17 \( 1 + (-0.442 + 0.896i)T \)
19 \( 1 + (0.347 + 0.937i)T \)
23 \( 1 + (-0.316 + 0.948i)T \)
29 \( 1 + (-0.592 + 0.805i)T \)
31 \( 1 + (-0.997 - 0.0714i)T \)
37 \( 1 + (-0.847 - 0.530i)T \)
41 \( 1 + (0.0682 - 0.997i)T \)
43 \( 1 + (0.927 - 0.374i)T \)
47 \( 1 + (-0.861 + 0.508i)T \)
53 \( 1 + (-0.877 + 0.480i)T \)
59 \( 1 + (-0.171 - 0.985i)T \)
61 \( 1 + (0.898 - 0.439i)T \)
67 \( 1 + (-0.560 - 0.828i)T \)
71 \( 1 + (0.389 - 0.921i)T \)
73 \( 1 + (-0.877 - 0.480i)T \)
79 \( 1 + (-0.581 - 0.813i)T \)
83 \( 1 + (-0.359 + 0.933i)T \)
89 \( 1 + (-0.436 + 0.899i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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

−20.90571733151568433359403503338, −20.30697951068089326262355761870, −19.72602477648678350278415491066, −18.80439753857533684767102727056, −18.06246224770061074908556599025, −17.59978243427958303178408734348, −16.45951235771309304301564546140, −15.717390694252348420583809106784, −14.983937377673773884125607319840, −14.335366073676448076088880649048, −13.088189620323162184472859551211, −12.579837143114650210719096053145, −11.54678431424204031049087818918, −10.119980470751102680209181317098, −9.72942884466218114152303766801, −9.12142497815947911878982693079, −8.4176866729069360659049485681, −7.31615871193623529046194217124, −6.73433378302887598976145875431, −5.54687300032438074097263374082, −4.63474830306119041853325308897, −2.760726457731094920103421764043, −2.508459764014384659672960230997, −1.5814893072909866694342315792, −0.10258362612697309190414025967, 1.420488595405196625143766789121, 2.06033976734140567978117577906, 3.20711323996533399268590134417, 3.77255068450271583835601509243, 5.525747925124707933120840060433, 6.54074047979867840870947607882, 7.32454347300529716805576659867, 8.00035277190277764906170316935, 9.158860804776288795810405053979, 9.521700371672968384209707517429, 10.48011432011363612823817641574, 10.8786019885942146480606895415, 12.394200034932906397797474984444, 13.140292274033145276243661757, 14.08155306895917172795952907538, 14.57537027621620302842639425576, 15.72910311064633259017966544007, 16.48205038056167604649259118423, 17.216234901038030709048194810942, 17.99184912436906037194497678634, 19.02247965408826611096683437865, 19.29770382409051002297805074765, 20.224372258866497028416160991596, 20.86894814325498032612831647872, 21.66749305446152589069648336894

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