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Properties

Label 1-2303-2303.281-r1-0-0
Degree $1$
Conductor $2303$
Sign $0.404 - 0.914i$
Analytic cond. $247.491$
Root an. cond. $247.491$
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  + (−0.900 + 0.433i)2-s + (−0.222 + 0.974i)3-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + (−0.222 − 0.974i)6-s + (−0.222 + 0.974i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (0.900 − 0.433i)11-s + (0.623 + 0.781i)12-s + (0.900 − 0.433i)13-s + (0.900 + 0.433i)15-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s + 18-s − 19-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (−0.222 + 0.974i)3-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + (−0.222 − 0.974i)6-s + (−0.222 + 0.974i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (0.900 − 0.433i)11-s + (0.623 + 0.781i)12-s + (0.900 − 0.433i)13-s + (0.900 + 0.433i)15-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s + 18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2303\)    =    \(7^{2} \cdot 47\)
Sign: $0.404 - 0.914i$
Analytic conductor: \(247.491\)
Root analytic conductor: \(247.491\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2303} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2303,\ (1:\ ),\ 0.404 - 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7644435703 - 0.4975976215i\)
\(L(\frac12)\) \(\approx\) \(0.7644435703 - 0.4975976215i\)
\(L(1)\) \(\approx\) \(0.6829317525 + 0.1187070421i\)
\(L(1)\) \(\approx\) \(0.6829317525 + 0.1187070421i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.900 + 0.433i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (0.623 + 0.781i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.623 + 0.781i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 - T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 - T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + T \)
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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

−19.41956917180007768877040817206, −18.67679739288227938924089176428, −18.24851703555040159942420731755, −17.82134113795869258460985935794, −16.713004988631238376315301457854, −16.556527974828816074116534647635, −15.24547368252857881402960480762, −14.41999527535568375709279626207, −13.82107892047609708608327304021, −12.793754011806742488058588638688, −12.251106650714253352145752786958, −11.33685021126143290553251004250, −11.023863838013678607900488439463, −10.14183895469788263820885041333, −9.220199138134080485868550746634, −8.59112473375367060834316794981, −7.5911404295972950291883072158, −7.05520898725509500805445596172, −6.41900766441700081988598167212, −5.77463759171141363811609810392, −4.1473915554602165317675448264, −3.32420394280968062605478540910, −2.33418328692735688969678270267, −1.79238380909047950063149256215, −0.85597770290885590461184313161, 0.26976788772357063452079610691, 1.1531617760877922792784376204, 2.04746825335380868044335583763, 3.52196425576459615130175823613, 4.150621812948483780892050734873, 5.26283537531244213575758061817, 5.936501630919253288534563845564, 6.3195763786682758931014955318, 7.79787485168278877436301293086, 8.436181802738497273934902414165, 9.01008339563993561691870044193, 9.65445079896002775662091312923, 10.33947343112802512626625355926, 11.16374432418894260466999890944, 11.72072836784383891791247849777, 12.72369852128269797365522516361, 13.72249874920147713068259206423, 14.585394775716736461078441757419, 15.21550862516661855424428662481, 15.96402984028121163068671102240, 16.55831496172411595695993287517, 17.06436236350627825608045747763, 17.562730421994283249599503411968, 18.48511385755417105492606229476, 19.5417689798647478452434374298

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