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Properties

Label 1-833-833.61-r0-0-0
Degree $1$
Conductor $833$
Sign $-0.600 - 0.799i$
Analytic cond. $3.86843$
Root an. cond. $3.86843$
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  + (−0.916 + 0.399i)2-s + (−0.949 + 0.312i)3-s + (0.680 − 0.733i)4-s + (−0.978 − 0.204i)5-s + (0.745 − 0.666i)6-s + (−0.330 + 0.943i)8-s + (0.804 − 0.593i)9-s + (0.978 − 0.204i)10-s + (0.240 − 0.970i)11-s + (−0.416 + 0.908i)12-s + (−0.781 + 0.623i)13-s + (0.993 − 0.111i)15-s + (−0.0747 − 0.997i)16-s + (−0.5 + 0.866i)18-s + (0.258 − 0.965i)19-s + (−0.815 + 0.578i)20-s + ⋯
L(s)  = 1  + (−0.916 + 0.399i)2-s + (−0.949 + 0.312i)3-s + (0.680 − 0.733i)4-s + (−0.978 − 0.204i)5-s + (0.745 − 0.666i)6-s + (−0.330 + 0.943i)8-s + (0.804 − 0.593i)9-s + (0.978 − 0.204i)10-s + (0.240 − 0.970i)11-s + (−0.416 + 0.908i)12-s + (−0.781 + 0.623i)13-s + (0.993 − 0.111i)15-s + (−0.0747 − 0.997i)16-s + (−0.5 + 0.866i)18-s + (0.258 − 0.965i)19-s + (−0.815 + 0.578i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $-0.600 - 0.799i$
Analytic conductor: \(3.86843\)
Root analytic conductor: \(3.86843\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 833,\ (0:\ ),\ -0.600 - 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08411104140 - 0.1683021896i\)
\(L(\frac12)\) \(\approx\) \(0.08411104140 - 0.1683021896i\)
\(L(1)\) \(\approx\) \(0.3974532808 + 0.01423206071i\)
\(L(1)\) \(\approx\) \(0.3974532808 + 0.01423206071i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.916 + 0.399i)T \)
3 \( 1 + (-0.949 + 0.312i)T \)
5 \( 1 + (-0.978 - 0.204i)T \)
11 \( 1 + (0.240 - 0.970i)T \)
13 \( 1 + (-0.781 + 0.623i)T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (0.0933 - 0.995i)T \)
29 \( 1 + (0.578 + 0.815i)T \)
31 \( 1 + (-0.991 - 0.130i)T \)
37 \( 1 + (-0.908 - 0.416i)T \)
41 \( 1 + (0.998 - 0.0560i)T \)
43 \( 1 + (0.943 - 0.330i)T \)
47 \( 1 + (0.930 + 0.365i)T \)
53 \( 1 + (0.0373 + 0.999i)T \)
59 \( 1 + (-0.185 + 0.982i)T \)
61 \( 1 + (-0.937 + 0.347i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.985 + 0.167i)T \)
73 \( 1 + (-0.693 - 0.720i)T \)
79 \( 1 + (0.991 - 0.130i)T \)
83 \( 1 + (-0.993 + 0.111i)T \)
89 \( 1 + (-0.149 + 0.988i)T \)
97 \( 1 + (0.923 - 0.382i)T \)
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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

−22.553409629417757764695405488736, −21.67091234973864571594613818166, −20.62510678171040720341784836306, −19.82552946994330230217723644195, −19.179293548914958478980251666170, −18.45804205703431901945726040361, −17.56559691133546182110455327878, −17.169812958555695458464377783146, −16.08395929001246352692478269817, −15.581489380853769246026959240888, −14.587587215851409852726067169569, −13.01274089567625575119524407137, −12.23638492307755987205536001216, −11.88984542830539917129845425416, −10.95922630875109233460501820458, −10.21984478052872775458376400995, −9.46086044577357396542250600801, −8.08325036177074597323322437929, −7.46826523148173306942382928229, −6.915272534381995617264921357323, −5.70866483935695141580047256513, −4.503748972901171146138989156293, −3.54637202755303666846425087611, −2.26673358878264759006114026590, −1.154052272428551980783246246, 0.17002859229183873804259742123, 1.16543550617846499553035334323, 2.790902998509038334756519459310, 4.16203718762717660832447887538, 5.03528932610618773747654219563, 5.98810443160960627477878208874, 6.97298245495857078084339004151, 7.50996533528621484295364997719, 8.844717619238967103572304320331, 9.215984383777470968338433673447, 10.64414725765523024926050772049, 10.951305681511109417672018229285, 11.896534044688656817151797926577, 12.472355510093350620540679008532, 14.04847506183551544198963164328, 14.95494373064565497970480121382, 15.78633652771842205202563878574, 16.37801440714010052728776692306, 16.88556073259639254813643785652, 17.78085219827761290192892626391, 18.65295801551021872641772368059, 19.30728705745950412602652792634, 20.03853247087912526207162988323, 21.030863380241109114365288477684, 21.96048050369612737076028376714

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