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Properties

Label 1-967-967.109-r1-0-0
Degree $1$
Conductor $967$
Sign $0.0546 - 0.998i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.165 + 0.986i)2-s + (0.822 − 0.568i)3-s + (−0.945 + 0.325i)4-s + (0.977 − 0.212i)5-s + (0.696 + 0.717i)6-s + (−0.241 − 0.970i)7-s + (−0.477 − 0.878i)8-s + (0.353 − 0.935i)9-s + (0.371 + 0.928i)10-s + (−0.511 − 0.859i)11-s + (−0.592 + 0.805i)12-s + (0.511 − 0.859i)13-s + (0.917 − 0.398i)14-s + (0.682 − 0.730i)15-s + (0.787 − 0.615i)16-s + (0.996 − 0.0779i)17-s + ⋯
L(s)  = 1  + (0.165 + 0.986i)2-s + (0.822 − 0.568i)3-s + (−0.945 + 0.325i)4-s + (0.977 − 0.212i)5-s + (0.696 + 0.717i)6-s + (−0.241 − 0.970i)7-s + (−0.477 − 0.878i)8-s + (0.353 − 0.935i)9-s + (0.371 + 0.928i)10-s + (−0.511 − 0.859i)11-s + (−0.592 + 0.805i)12-s + (0.511 − 0.859i)13-s + (0.917 − 0.398i)14-s + (0.682 − 0.730i)15-s + (0.787 − 0.615i)16-s + (0.996 − 0.0779i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0546 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0546 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.0546 - 0.998i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.0546 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.103780828 - 1.991703415i\)
\(L(\frac12)\) \(\approx\) \(2.103780828 - 1.991703415i\)
\(L(1)\) \(\approx\) \(1.540831750 - 0.1313875026i\)
\(L(1)\) \(\approx\) \(1.540831750 - 0.1313875026i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.165 + 0.986i)T \)
3 \( 1 + (0.822 - 0.568i)T \)
5 \( 1 + (0.977 - 0.212i)T \)
7 \( 1 + (-0.241 - 0.970i)T \)
11 \( 1 + (-0.511 - 0.859i)T \)
13 \( 1 + (0.511 - 0.859i)T \)
17 \( 1 + (0.996 - 0.0779i)T \)
19 \( 1 + (-0.494 + 0.869i)T \)
23 \( 1 + (-0.527 - 0.849i)T \)
29 \( 1 + (0.724 + 0.689i)T \)
31 \( 1 + (0.389 - 0.921i)T \)
37 \( 1 + (-0.833 + 0.552i)T \)
41 \( 1 + (-0.854 + 0.519i)T \)
43 \( 1 + (0.145 + 0.989i)T \)
47 \( 1 + (0.297 - 0.954i)T \)
53 \( 1 + (-0.334 + 0.942i)T \)
59 \( 1 + (-0.107 + 0.994i)T \)
61 \( 1 + (0.854 - 0.519i)T \)
67 \( 1 + (-0.389 - 0.921i)T \)
71 \( 1 + (0.165 - 0.986i)T \)
73 \( 1 + (-0.334 - 0.942i)T \)
79 \( 1 + (-0.993 + 0.116i)T \)
83 \( 1 + (0.126 - 0.991i)T \)
89 \( 1 + (-0.389 - 0.921i)T \)
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

−21.60776494052006613914659921242, −21.00182064549822154857290986836, −20.48309163011689181377664764864, −19.17790547381413828525312106173, −19.05032976797256940503208632228, −17.963096809094251021370929271095, −17.329453293877304032161871731732, −15.95420122712850786279887820502, −15.26543622248946077803483994923, −14.324022393231554291648903142504, −13.83032032057733418561278117036, −12.98314744085851892803837646154, −12.22751984355689264783399065984, −11.1876572819461048161096719126, −10.13585634744740212353955702612, −9.80972621453179707460511388599, −8.96275699000404454706425916604, −8.36431089465163720773784303281, −6.91160920653025806791617648948, −5.631985601471789755003635042, −5.00776143461579697187851524618, −3.94138543095324765433250954035, −2.92962448750724433331560378923, −2.250209925948735141943568828881, −1.56714042896349959840228464856, 0.49536045232031759416227584656, 1.35624115868500908188301749537, 2.932929760450441560457433659914, 3.5970580626812033701795665786, 4.79443305209934646485544662162, 5.99223507968579550143696873459, 6.35433504260974616986985174190, 7.52642865782309314272527204792, 8.189194304386387303261006067156, 8.800360206830250617232105874400, 10.026288173207634741728158955111, 10.35550675992115891049766694788, 12.23788357489859520052683097966, 13.01248070686246551071004257173, 13.548478003640084864636337528694, 14.11583954816446474195685082969, 14.77563475794544829731898153991, 15.891287768037282940245231574, 16.66219294282268895514726967742, 17.28274931932744167286002204773, 18.381891411746204118083053952, 18.576488109557632384156419122844, 19.79306793712846930341587487206, 20.7963334396585327705817281291, 21.1594249787490652113312588523

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