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Properties

Label 1-967-967.95-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.182 - 0.983i$
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  + (−0.724 − 0.689i)2-s + (−0.442 − 0.896i)3-s + (0.0487 + 0.998i)4-s + (−0.696 − 0.717i)5-s + (−0.297 + 0.954i)6-s + (0.981 + 0.193i)7-s + (0.653 − 0.756i)8-s + (−0.608 + 0.793i)9-s + (0.00975 + 0.999i)10-s + (0.909 + 0.416i)11-s + (0.874 − 0.485i)12-s + (0.909 − 0.416i)13-s + (−0.576 − 0.816i)14-s + (−0.334 + 0.942i)15-s + (−0.995 + 0.0974i)16-s + (−0.844 + 0.536i)17-s + ⋯
L(s)  = 1  + (−0.724 − 0.689i)2-s + (−0.442 − 0.896i)3-s + (0.0487 + 0.998i)4-s + (−0.696 − 0.717i)5-s + (−0.297 + 0.954i)6-s + (0.981 + 0.193i)7-s + (0.653 − 0.756i)8-s + (−0.608 + 0.793i)9-s + (0.00975 + 0.999i)10-s + (0.909 + 0.416i)11-s + (0.874 − 0.485i)12-s + (0.909 − 0.416i)13-s + (−0.576 − 0.816i)14-s + (−0.334 + 0.942i)15-s + (−0.995 + 0.0974i)16-s + (−0.844 + 0.536i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.182 - 0.983i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.182 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5389115724 - 0.6484729315i\)
\(L(\frac12)\) \(\approx\) \(0.5389115724 - 0.6484729315i\)
\(L(1)\) \(\approx\) \(0.5700401868 - 0.3857576279i\)
\(L(1)\) \(\approx\) \(0.5700401868 - 0.3857576279i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.724 - 0.689i)T \)
3 \( 1 + (-0.442 - 0.896i)T \)
5 \( 1 + (-0.696 - 0.717i)T \)
7 \( 1 + (0.981 + 0.193i)T \)
11 \( 1 + (0.909 + 0.416i)T \)
13 \( 1 + (0.909 - 0.416i)T \)
17 \( 1 + (-0.844 + 0.536i)T \)
19 \( 1 + (-0.977 - 0.212i)T \)
23 \( 1 + (-0.477 - 0.878i)T \)
29 \( 1 + (-0.999 - 0.0195i)T \)
31 \( 1 + (0.592 + 0.805i)T \)
37 \( 1 + (0.892 + 0.451i)T \)
41 \( 1 + (0.682 - 0.730i)T \)
43 \( 1 + (0.993 + 0.116i)T \)
47 \( 1 + (0.527 + 0.849i)T \)
53 \( 1 + (0.962 - 0.269i)T \)
59 \( 1 + (0.389 + 0.921i)T \)
61 \( 1 + (0.682 - 0.730i)T \)
67 \( 1 + (0.592 - 0.805i)T \)
71 \( 1 + (-0.724 + 0.689i)T \)
73 \( 1 + (0.962 + 0.269i)T \)
79 \( 1 + (-0.750 - 0.660i)T \)
83 \( 1 + (-0.864 - 0.502i)T \)
89 \( 1 + (0.592 - 0.805i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−22.100593290995442207816404618223, −21.17732283771176895453446252804, −20.28120173377377734879740089559, −19.55527968576430757512890189085, −18.617496394612449685201127458353, −17.919644458244486888073560240181, −17.18438264589268456953603340738, −16.43906584908103498744269611873, −15.70567085891415571614876056090, −14.98330553986465987939054704489, −14.45681121039774367300225296164, −13.622289855694560296260748116116, −11.740507278083219251192239661097, −11.204451183791702333106972640232, −10.88027278719576263672122736856, −9.75504960120433762842585031858, −8.8922181754415035225860546154, −8.21673083247852963974201664787, −7.199903619528222446603909612789, −6.32730480904055236945428120852, −5.623644810883162115095946620728, −4.255496170917498320393255899834, −3.98340855347599231378508756998, −2.253881768548925899806591820898, −0.833678810730381836629148157228, 0.75968640661902128126806297683, 1.57620716452512136597560971900, 2.40569730603480087806219032828, 3.99408943777925357639783585220, 4.55845869082532587713468028133, 5.94489344893815726828107658713, 6.9527116177293030003660849268, 7.86857501610220582584710752658, 8.52777610908246187621507385630, 8.98284480887601351666224029548, 10.57563198203512062023921337968, 11.18938346398582080671509503040, 11.79256437199738098555560034178, 12.60285635049461667487698290612, 13.05904523073353557602860752861, 14.227938973013058663535020231054, 15.35437984515346471762417879381, 16.313120323647229306631869107184, 17.22039711262226734726968252310, 17.55322909472850915106094864981, 18.43178963213561740290256849028, 19.16612573523771826096534245718, 19.92270514049243470949895036420, 20.45398775332771389011942907531, 21.32993028299815148737929083981

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