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Properties

Label 1-967-967.137-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

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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: $\chi_{967} (137, \cdot )$
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 \( 1 + (-0.222 + 0.974i)T \)
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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

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

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