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Properties

Label 1-967-967.462-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: $\chi_{967} (462, \cdot )$
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

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

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