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Properties

Label 1-967-967.186-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.753 - 0.656i$
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  + (−0.511 + 0.859i)2-s + (−0.787 − 0.615i)3-s + (−0.477 − 0.878i)4-s + (−0.316 − 0.948i)5-s + (0.932 − 0.362i)6-s + (0.407 − 0.913i)7-s + (0.999 + 0.0390i)8-s + (0.241 + 0.970i)9-s + (0.977 + 0.212i)10-s + (−0.999 + 0.0195i)11-s + (−0.165 + 0.986i)12-s + (0.999 + 0.0195i)13-s + (0.576 + 0.816i)14-s + (−0.334 + 0.942i)15-s + (−0.544 + 0.838i)16-s + (0.993 − 0.116i)17-s + ⋯
L(s)  = 1  + (−0.511 + 0.859i)2-s + (−0.787 − 0.615i)3-s + (−0.477 − 0.878i)4-s + (−0.316 − 0.948i)5-s + (0.932 − 0.362i)6-s + (0.407 − 0.913i)7-s + (0.999 + 0.0390i)8-s + (0.241 + 0.970i)9-s + (0.977 + 0.212i)10-s + (−0.999 + 0.0195i)11-s + (−0.165 + 0.986i)12-s + (0.999 + 0.0195i)13-s + (0.576 + 0.816i)14-s + (−0.334 + 0.942i)15-s + (−0.544 + 0.838i)16-s + (0.993 − 0.116i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.753 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.753 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.753 - 0.656i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.753 - 0.656i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3550759894 - 0.9481172386i\)
\(L(\frac12)\) \(\approx\) \(0.3550759894 - 0.9481172386i\)
\(L(1)\) \(\approx\) \(0.6204544472 - 0.2017053908i\)
\(L(1)\) \(\approx\) \(0.6204544472 - 0.2017053908i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.511 + 0.859i)T \)
3 \( 1 + (-0.787 - 0.615i)T \)
5 \( 1 + (-0.316 - 0.948i)T \)
7 \( 1 + (0.407 - 0.913i)T \)
11 \( 1 + (-0.999 + 0.0195i)T \)
13 \( 1 + (0.999 + 0.0195i)T \)
17 \( 1 + (0.993 - 0.116i)T \)
19 \( 1 + (-0.00975 - 0.999i)T \)
23 \( 1 + (-0.0487 - 0.998i)T \)
29 \( 1 + (-0.909 + 0.416i)T \)
31 \( 1 + (-0.184 - 0.982i)T \)
37 \( 1 + (0.638 - 0.769i)T \)
41 \( 1 + (-0.682 + 0.730i)T \)
43 \( 1 + (0.844 + 0.536i)T \)
47 \( 1 + (0.945 - 0.325i)T \)
53 \( 1 + (0.962 - 0.269i)T \)
59 \( 1 + (0.811 - 0.584i)T \)
61 \( 1 + (0.682 - 0.730i)T \)
67 \( 1 + (0.184 - 0.982i)T \)
71 \( 1 + (-0.511 - 0.859i)T \)
73 \( 1 + (0.962 + 0.269i)T \)
79 \( 1 + (0.984 - 0.174i)T \)
83 \( 1 + (0.560 + 0.828i)T \)
89 \( 1 + (0.184 - 0.982i)T \)
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

−21.8095767791617759175394791244, −20.93463860519023788274854189485, −20.72862060189981521742819760385, −19.21527159245445854461497898552, −18.54370960838967489270218523542, −18.21300514635011570511203803496, −17.38673857547839910960944394863, −16.33828713971501355468862874797, −15.66785280142029272730404534676, −14.895009747970692263509548199854, −13.832791876312684188784914956530, −12.6999991454649511548841658932, −11.87045841073668394351485731771, −11.391921986454018077405016026620, −10.51332447351381292487339638751, −10.11626543914229105986222206524, −9.02795353107026628565004176424, −8.101172229462625860723393235647, −7.29518322956013088712566625578, −5.89238092927174094148163680416, −5.32490627612745810685518000714, −3.936406373236536815761854987614, −3.3676168928633554706598667100, −2.2869861985434647351458102632, −1.061007679819854253427654943193, 0.472785433476995001440004537481, 0.79734127319187657353891396568, 1.96859663118154707697773861017, 3.98659744483828361475547426026, 4.90842643217395918895959868344, 5.48979599378386115121328793533, 6.44852643544104175639759630240, 7.520574268564488280988619970820, 7.86284453203092239535561976874, 8.75301846111472424308146998687, 9.891688099061833212120664144426, 10.83065925813064019421482452852, 11.34922430305053350600897924689, 12.727393871745700072985535484900, 13.24083584506585094922543662601, 13.970045935921395860095202696394, 15.143707528713091694946203006960, 16.11938809564133120458198914249, 16.56136745873732997626631720193, 17.15006524012017420178626449984, 18.09581886197203965046001412361, 18.52955098931140269838024225573, 19.51400246082590158632962062259, 20.36367161777956043245154135904, 21.081314229131349024656175029333

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