↑

Properties

Label 1-967-967.926-r0-0-0
Degree $1$
Conductor $967$
Sign $0.805 - 0.592i$
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  + (−0.576 − 0.816i)2-s + (−0.0682 + 0.997i)3-s + (−0.334 + 0.942i)4-s + (−0.775 − 0.631i)5-s + (0.854 − 0.519i)6-s + (0.203 − 0.979i)7-s + (0.962 − 0.269i)8-s + (−0.990 − 0.136i)9-s + (−0.0682 + 0.997i)10-s + (−0.990 − 0.136i)11-s + (−0.917 − 0.398i)12-s + (−0.990 + 0.136i)13-s + (−0.917 + 0.398i)14-s + (0.682 − 0.730i)15-s + (−0.775 − 0.631i)16-s + (0.682 + 0.730i)17-s + ⋯
L(s)  = 1  + (−0.576 − 0.816i)2-s + (−0.0682 + 0.997i)3-s + (−0.334 + 0.942i)4-s + (−0.775 − 0.631i)5-s + (0.854 − 0.519i)6-s + (0.203 − 0.979i)7-s + (0.962 − 0.269i)8-s + (−0.990 − 0.136i)9-s + (−0.0682 + 0.997i)10-s + (−0.990 − 0.136i)11-s + (−0.917 − 0.398i)12-s + (−0.990 + 0.136i)13-s + (−0.917 + 0.398i)14-s + (0.682 − 0.730i)15-s + (−0.775 − 0.631i)16-s + (0.682 + 0.730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (926, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.805 - 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5450293074 - 0.1788145691i\)
\(L(\frac12)\) \(\approx\) \(0.5450293074 - 0.1788145691i\)
\(L(1)\) \(\approx\) \(0.5652763275 - 0.1018123991i\)
\(L(1)\) \(\approx\) \(0.5652763275 - 0.1018123991i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.576 - 0.816i)T \)
3 \( 1 + (-0.0682 + 0.997i)T \)
5 \( 1 + (-0.775 - 0.631i)T \)
7 \( 1 + (0.203 - 0.979i)T \)
11 \( 1 + (-0.990 - 0.136i)T \)
13 \( 1 + (-0.990 + 0.136i)T \)
17 \( 1 + (0.682 + 0.730i)T \)
19 \( 1 + (-0.0682 + 0.997i)T \)
23 \( 1 + (-0.334 + 0.942i)T \)
29 \( 1 + (-0.990 + 0.136i)T \)
31 \( 1 + (0.962 - 0.269i)T \)
37 \( 1 + (-0.990 + 0.136i)T \)
41 \( 1 + (0.854 - 0.519i)T \)
43 \( 1 + (0.682 - 0.730i)T \)
47 \( 1 + (0.682 - 0.730i)T \)
53 \( 1 + (-0.334 + 0.942i)T \)
59 \( 1 + (-0.334 - 0.942i)T \)
61 \( 1 + (0.854 - 0.519i)T \)
67 \( 1 + (0.962 + 0.269i)T \)
71 \( 1 + (-0.576 + 0.816i)T \)
73 \( 1 + (-0.334 - 0.942i)T \)
79 \( 1 + (-0.334 - 0.942i)T \)
83 \( 1 + (0.854 - 0.519i)T \)
89 \( 1 + (0.962 + 0.269i)T \)
97 \( 1 + T \)
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

−22.345713275157912913385499439860, −20.89136944336210133622631292845, −19.802527318171979057954099444, −19.14974623443044991127953110080, −18.609348019363729381259199841291, −18.00967381803771167058087465441, −17.3615164321012732050471942680, −16.19391030933087701386358809426, −15.55520193456073954849695474450, −14.70863981565219945650761294351, −14.218317725396534560555877360256, −13.04005315457471553345087911677, −12.206771462824521702876738429968, −11.40128639591171726192760151255, −10.54483951127398108465384527211, −9.45881058614081519936901568669, −8.46917500841387411011460638465, −7.76324544710284865538021133176, −7.265344715369441329787225779149, −6.38383745006388684890510436409, −5.44100285211076360898909515528, −4.69894874470987359359844801485, −2.802337701666379382868269507638, −2.28517045705880305963344797838, −0.62735023381440792910101603952, 0.53806703475576652893444610433, 1.94019924575288205376283847776, 3.342666948351649136129697146454, 3.88363143354843374678094812288, 4.69554989203157266889372075333, 5.56302084021328217392161811779, 7.468848027414295263513657483826, 7.8680136359147311403695318580, 8.75621701370214692479432161125, 9.77564431754204419066781888330, 10.32772696488035697538834068792, 11.02113912774976784078172647873, 11.91771287862456040811003308498, 12.56831173910155039691132557404, 13.61352381537100173696687953838, 14.53150398021994728879319830375, 15.62822077039533493115178070898, 16.27574770915695496602169753643, 17.16188684956248630654734569664, 17.32595690956319978530564508124, 18.88458572275346245432310029126, 19.37942642424554765857022425852, 20.39869939857794164947970725192, 20.62837382348000656872459121919, 21.38544188494972841599173911882

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