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Properties

Label 1-1128-1128.845-r0-0-0
Degree $1$
Conductor $1128$
Sign $1$
Analytic cond. $5.23840$
Root an. cond. $5.23840$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 11-s + 13-s − 17-s + 19-s + 23-s + 25-s − 29-s − 31-s − 35-s − 37-s + 41-s + 43-s + 49-s + 53-s + 55-s + 59-s − 61-s − 65-s + 67-s − 71-s − 73-s − 77-s + 79-s + 83-s + 85-s + ⋯
L(s)  = 1  − 5-s + 7-s − 11-s + 13-s − 17-s + 19-s + 23-s + 25-s − 29-s − 31-s − 35-s − 37-s + 41-s + 43-s + 49-s + 53-s + 55-s + 59-s − 61-s − 65-s + 67-s − 71-s − 73-s − 77-s + 79-s + 83-s + 85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1128 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1128 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1128\)    =    \(2^{3} \cdot 3 \cdot 47\)
Sign: $1$
Analytic conductor: \(5.23840\)
Root analytic conductor: \(5.23840\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1128} (845, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1128,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.326343622\)
\(L(\frac12)\) \(\approx\) \(1.326343622\)
\(L(1)\) \(\approx\) \(1.007064358\)
\(L(1)\) \(\approx\) \(1.007064358\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
47 \( 1 \)
good5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 - T \)
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

−21.03608063640034083587896789635, −20.67902075965776966420975846664, −19.91177514056294106416721101308, −18.92173991748948076561582776456, −18.26803384257408599554971256945, −17.68143674696327130301816668953, −16.54234427644070831405034164303, −15.78606981859555053411798003791, −15.26457121248463436876246817237, −14.42204360171607318014563511029, −13.44864616067645723937319997455, −12.759253096268233864522126701898, −11.69155189733617367620611734851, −11.053048163434881254969634479262, −10.63593298995419641766206528908, −9.11591773512552504905050278929, −8.53671021754386496623655653449, −7.60271991897966211425181690353, −7.15144595908531724034844899886, −5.72748874521571355832738890385, −4.96935197147393370604083924652, −4.08717867253428377984713958016, −3.199686500146097602894631806696, −2.0428857947310067756122789834, −0.82140231019765282213109299525, 0.82140231019765282213109299525, 2.0428857947310067756122789834, 3.199686500146097602894631806696, 4.08717867253428377984713958016, 4.96935197147393370604083924652, 5.72748874521571355832738890385, 7.15144595908531724034844899886, 7.60271991897966211425181690353, 8.53671021754386496623655653449, 9.11591773512552504905050278929, 10.63593298995419641766206528908, 11.053048163434881254969634479262, 11.69155189733617367620611734851, 12.759253096268233864522126701898, 13.44864616067645723937319997455, 14.42204360171607318014563511029, 15.26457121248463436876246817237, 15.78606981859555053411798003791, 16.54234427644070831405034164303, 17.68143674696327130301816668953, 18.26803384257408599554971256945, 18.92173991748948076561582776456, 19.91177514056294106416721101308, 20.67902075965776966420975846664, 21.03608063640034083587896789635

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