↑

Properties

Label 1-967-967.120-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.244 + 0.969i$
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  + (−0.964 − 0.263i)2-s + (−0.511 − 0.859i)3-s + (0.861 + 0.508i)4-s + (−0.837 + 0.547i)5-s + (0.266 + 0.963i)6-s + (−0.465 + 0.884i)7-s + (−0.696 − 0.717i)8-s + (−0.477 + 0.878i)9-s + (0.951 − 0.307i)10-s + (0.389 + 0.921i)11-s + (−0.00325 − 0.999i)12-s + (0.602 + 0.797i)13-s + (0.682 − 0.730i)14-s + (0.898 + 0.439i)15-s + (0.483 + 0.875i)16-s + (0.737 + 0.675i)17-s + ⋯
L(s)  = 1  + (−0.964 − 0.263i)2-s + (−0.511 − 0.859i)3-s + (0.861 + 0.508i)4-s + (−0.837 + 0.547i)5-s + (0.266 + 0.963i)6-s + (−0.465 + 0.884i)7-s + (−0.696 − 0.717i)8-s + (−0.477 + 0.878i)9-s + (0.951 − 0.307i)10-s + (0.389 + 0.921i)11-s + (−0.00325 − 0.999i)12-s + (0.602 + 0.797i)13-s + (0.682 − 0.730i)14-s + (0.898 + 0.439i)15-s + (0.483 + 0.875i)16-s + (0.737 + 0.675i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.244 + 0.969i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (120, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.244 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2845702933 + 0.3651142088i\)
\(L(\frac12)\) \(\approx\) \(0.2845702933 + 0.3651142088i\)
\(L(1)\) \(\approx\) \(0.4960128576 + 0.05156658779i\)
\(L(1)\) \(\approx\) \(0.4960128576 + 0.05156658779i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.964 - 0.263i)T \)
3 \( 1 + (-0.511 - 0.859i)T \)
5 \( 1 + (-0.837 + 0.547i)T \)
7 \( 1 + (-0.465 + 0.884i)T \)
11 \( 1 + (0.389 + 0.921i)T \)
13 \( 1 + (0.602 + 0.797i)T \)
17 \( 1 + (0.737 + 0.675i)T \)
19 \( 1 + (0.0617 - 0.998i)T \)
23 \( 1 + (-0.977 + 0.212i)T \)
29 \( 1 + (0.811 + 0.584i)T \)
31 \( 1 + (0.795 + 0.605i)T \)
37 \( 1 + (-0.197 - 0.980i)T \)
41 \( 1 + (0.460 - 0.887i)T \)
43 \( 1 + (-0.0812 + 0.996i)T \)
47 \( 1 + (-0.999 - 0.00650i)T \)
53 \( 1 + (-0.158 + 0.987i)T \)
59 \( 1 + (-0.285 + 0.958i)T \)
61 \( 1 + (0.538 + 0.842i)T \)
67 \( 1 + (0.126 + 0.991i)T \)
71 \( 1 + (0.710 + 0.703i)T \)
73 \( 1 + (-0.158 - 0.987i)T \)
79 \( 1 + (0.997 + 0.0649i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (-0.922 - 0.386i)T \)
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

−21.18684038011420346434680195563, −20.54846656287924796668442521038, −20.03343514140417219547634249786, −19.166492953361486931068344095740, −18.367781270281189053608647047808, −17.270672807548795088361812040951, −16.720007670113814138687203720792, −16.06831152300077376995678652052, −15.74055754253888405957906869990, −14.64759798940347853470304216961, −13.73451344448106721758467819085, −12.32997937834885714447406819113, −11.62968816204789330293226379811, −10.925386830675749193975597258073, −10.04904646849959762339149314060, −9.56702908394740951275473467910, −8.19933999904929835398720364389, −8.08706336299620883569272429926, −6.62536811217580718561438507829, −5.95960065280403399852981841278, −4.93801061470872726658812960285, −3.74923299073903900898945568688, −3.18048727978393892450664917250, −1.09081592567743097305349672731, −0.38861168066944064119804257260, 1.21281751780449012813466204695, 2.216953479841303635128288910482, 3.0830544405329769739422827605, 4.28518692583937258361181788769, 5.83454980427003596727793652721, 6.65943301678675828447535706441, 7.154554117952233177126867054208, 8.14080055459365601384451933384, 8.82399886121024933724793119943, 9.88964623949449347088606892785, 10.823187324530720782866775592570, 11.60916394005012133452646880061, 12.19257868615970688543348040527, 12.66450466951158017974036974563, 14.042324036715504823331773949006, 15.07490350389194141189137618016, 15.93227849658270280733540245830, 16.45035244371114217741299557198, 17.69292982455846602109749473725, 18.01084108123571771040355145674, 18.882243264827943478564819677, 19.49629304681529298248726798436, 19.81109705999851534510182280389, 21.214313648131665129343774320, 21.96124639343520357617343262397

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