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Properties

Label 1-967-967.877-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.289 - 0.957i$
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  + (−0.297 − 0.954i)2-s + (0.279 − 0.960i)3-s + (−0.822 + 0.568i)4-s + (−0.724 + 0.689i)5-s + (−0.999 + 0.0195i)6-s + (−0.750 + 0.660i)7-s + (0.787 + 0.615i)8-s + (−0.844 − 0.536i)9-s + (0.874 + 0.485i)10-s + (−0.945 + 0.325i)11-s + (0.316 + 0.948i)12-s + (−0.945 − 0.325i)13-s + (0.854 + 0.519i)14-s + (0.460 + 0.887i)15-s + (0.353 − 0.935i)16-s + (−0.407 − 0.913i)17-s + ⋯
L(s)  = 1  + (−0.297 − 0.954i)2-s + (0.279 − 0.960i)3-s + (−0.822 + 0.568i)4-s + (−0.724 + 0.689i)5-s + (−0.999 + 0.0195i)6-s + (−0.750 + 0.660i)7-s + (0.787 + 0.615i)8-s + (−0.844 − 0.536i)9-s + (0.874 + 0.485i)10-s + (−0.945 + 0.325i)11-s + (0.316 + 0.948i)12-s + (−0.945 − 0.325i)13-s + (0.854 + 0.519i)14-s + (0.460 + 0.887i)15-s + (0.353 − 0.935i)16-s + (−0.407 − 0.913i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.289 - 0.957i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (877, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3509492771 - 0.4729147151i\)
\(L(\frac12)\) \(\approx\) \(0.3509492771 - 0.4729147151i\)
\(L(1)\) \(\approx\) \(0.5313036083 - 0.3246695130i\)
\(L(1)\) \(\approx\) \(0.5313036083 - 0.3246695130i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.297 - 0.954i)T \)
3 \( 1 + (0.279 - 0.960i)T \)
5 \( 1 + (-0.724 + 0.689i)T \)
7 \( 1 + (-0.750 + 0.660i)T \)
11 \( 1 + (-0.945 + 0.325i)T \)
13 \( 1 + (-0.945 - 0.325i)T \)
17 \( 1 + (-0.407 - 0.913i)T \)
19 \( 1 + (0.165 + 0.986i)T \)
23 \( 1 + (0.737 + 0.675i)T \)
29 \( 1 + (0.527 - 0.849i)T \)
31 \( 1 + (0.00975 + 0.999i)T \)
37 \( 1 + (0.710 + 0.703i)T \)
41 \( 1 + (0.203 - 0.979i)T \)
43 \( 1 + (0.981 + 0.193i)T \)
47 \( 1 + (-0.799 - 0.600i)T \)
53 \( 1 + (-0.0682 + 0.997i)T \)
59 \( 1 + (-0.371 + 0.928i)T \)
61 \( 1 + (0.203 - 0.979i)T \)
67 \( 1 + (0.00975 - 0.999i)T \)
71 \( 1 + (-0.297 + 0.954i)T \)
73 \( 1 + (-0.0682 - 0.997i)T \)
79 \( 1 + (0.987 + 0.155i)T \)
83 \( 1 + (-0.638 - 0.769i)T \)
89 \( 1 + (0.00975 - 0.999i)T \)
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

−22.14009725652062747467834527054, −21.24998923913463174128948075057, −20.22131840826867195685017704906, −19.53702926669099483364396965718, −19.12396901349382916384147854673, −17.7256662334380012173229371321, −16.89637371542753600127194110884, −16.36753444084875060464639895388, −15.81339667196461306594583654261, −15.07486393798657403802898103637, −14.361599274883584063728604971925, −13.20835932223029107168693468517, −12.80705022353243395271041541031, −11.20725136084767711045778274734, −10.50651265819616123566598575274, −9.593243359250695971287443560543, −8.98039447174137474069622897846, −8.11812954524227399796006239506, −7.41686295516692798527319853981, −6.37475601907650146597235268672, −5.19067255232302311994696471290, −4.59252943088084427610040049416, −3.853336004433598765032484335637, −2.68016557231636078831189989438, −0.657942257297427419623900855569, 0.458300077111298910415041066374, 2.04099648177857104330756066035, 2.8600789944402870952525199628, 3.221453126500390089532905888827, 4.67360320888623147473750957937, 5.79584024796277006741645330209, 7.083026523736537834889595645366, 7.61940585694374233333144334374, 8.435957080697157303384657375894, 9.44032306728720551038743173403, 10.19940740400797007224630041048, 11.213713440845780514746263350242, 12.08632147051482774374033135362, 12.43219829843874012111866439357, 13.318787409174743996070217034105, 14.13297402659751972674418154913, 15.09700207475760062749936886526, 15.91737942588888407156095747997, 17.12768233980971780079453059127, 18.08812912034989912687458636230, 18.47553259143602856657302065900, 19.23973372312048546161262234254, 19.67827965515697523722306023002, 20.48753584437241663858325956330, 21.421470997350057807688217273713

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