↑

Properties

Label 1-967-967.77-r1-0-0
Degree $1$
Conductor $967$
Sign $0.895 + 0.444i$
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  + (−0.0552 − 0.998i)2-s + (−0.316 + 0.948i)3-s + (−0.993 + 0.110i)4-s + (0.997 − 0.0714i)5-s + (0.964 + 0.263i)6-s + (0.0812 + 0.996i)7-s + (0.165 + 0.986i)8-s + (−0.799 − 0.600i)9-s + (−0.126 − 0.991i)10-s + (0.763 − 0.646i)11-s + (0.209 − 0.977i)12-s + (0.941 − 0.337i)13-s + (0.990 − 0.136i)14-s + (−0.247 + 0.968i)15-s + (0.975 − 0.219i)16-s + (−0.477 + 0.878i)17-s + ⋯
L(s)  = 1  + (−0.0552 − 0.998i)2-s + (−0.316 + 0.948i)3-s + (−0.993 + 0.110i)4-s + (0.997 − 0.0714i)5-s + (0.964 + 0.263i)6-s + (0.0812 + 0.996i)7-s + (0.165 + 0.986i)8-s + (−0.799 − 0.600i)9-s + (−0.126 − 0.991i)10-s + (0.763 − 0.646i)11-s + (0.209 − 0.977i)12-s + (0.941 − 0.337i)13-s + (0.990 − 0.136i)14-s + (−0.247 + 0.968i)15-s + (0.975 − 0.219i)16-s + (−0.477 + 0.878i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.334403064 + 0.5470295883i\)
\(L(\frac12)\) \(\approx\) \(2.334403064 + 0.5470295883i\)
\(L(1)\) \(\approx\) \(1.189499588 + 0.02917873894i\)
\(L(1)\) \(\approx\) \(1.189499588 + 0.02917873894i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.0552 - 0.998i)T \)
3 \( 1 + (-0.316 + 0.948i)T \)
5 \( 1 + (0.997 - 0.0714i)T \)
7 \( 1 + (0.0812 + 0.996i)T \)
11 \( 1 + (0.763 - 0.646i)T \)
13 \( 1 + (0.941 - 0.337i)T \)
17 \( 1 + (-0.477 + 0.878i)T \)
19 \( 1 + (0.767 + 0.641i)T \)
23 \( 1 + (0.184 + 0.982i)T \)
29 \( 1 + (0.967 + 0.250i)T \)
31 \( 1 + (-0.132 + 0.991i)T \)
37 \( 1 + (0.322 - 0.946i)T \)
41 \( 1 + (0.334 - 0.942i)T \)
43 \( 1 + (-0.840 + 0.541i)T \)
47 \( 1 + (0.911 - 0.410i)T \)
53 \( 1 + (0.803 + 0.595i)T \)
59 \( 1 + (0.0357 - 0.999i)T \)
61 \( 1 + (-0.648 - 0.761i)T \)
67 \( 1 + (-0.924 - 0.380i)T \)
71 \( 1 + (0.892 - 0.451i)T \)
73 \( 1 + (0.803 - 0.595i)T \)
79 \( 1 + (0.465 - 0.884i)T \)
83 \( 1 + (-0.0422 + 0.999i)T \)
89 \( 1 + (0.791 - 0.610i)T \)
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

−21.82603520519128240287888764299, −20.56891166332975392585090989465, −19.887142291809831742130935203177, −18.73830033034014624456790110858, −18.071470098633188072233824568465, −17.60920862889402382556923421677, −16.80881868259688698178653558208, −16.34542098572583052026223817034, −15.0188723522538100204856460270, −14.10907789005441097001611888902, −13.5775035629162510593029003755, −13.21047285416272799386227277141, −12.011167296738876394533206096470, −10.998422475867581253298632562454, −9.984264963869542621448926344377, −9.16466336974398244989698530552, −8.288337242326252652562169866704, −7.17802062102731971569827836012, −6.738504092554973331420422782588, −6.11272504601839468465120143281, −5.021949355148201369565722639394, −4.231271453309312909836035388885, −2.70617179061121437770794979090, −1.32923376167737868468623730390, −0.71612597075658348977742464616, 0.96801421249736556464505873494, 1.915024357501995625697662416038, 3.13397845104014462763981309050, 3.724834807351577638544849119397, 4.97372372393404248649193404501, 5.686017399815871185138212300592, 6.25731152249693754848871515105, 8.30187609639731519331145297215, 8.98746966742304279590605915739, 9.41755748761485050862685470903, 10.46032212011956662583010931339, 10.9709913957839467940657102804, 11.89034399657523332867413653611, 12.581280298249246211855657250041, 13.70733452135874864186802076849, 14.2521284352773496041382359905, 15.23009011923860830980058528004, 16.17415602680802359384994758749, 17.06550784638131112723383205627, 17.83323858961818524581089221191, 18.31074307769432851662210445360, 19.433544056868991028372777201969, 20.177233388520851558946167945236, 21.18314485869364706294727200438, 21.483510632225718609684279664218

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