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Properties

Label 1-1148-1148.571-r0-0-0
Degree $1$
Conductor $1148$
Sign $-0.906 + 0.422i$
Analytic cond. $5.33128$
Root an. cond. $5.33128$
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  + (−0.258 + 0.965i)3-s + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)9-s + (0.258 − 0.965i)11-s + (0.707 + 0.707i)13-s + (0.707 − 0.707i)15-s + (−0.965 − 0.258i)17-s + (−0.258 − 0.965i)19-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s + (0.707 − 0.707i)27-s + (0.707 + 0.707i)29-s + (−0.5 − 0.866i)31-s + (0.866 + 0.5i)33-s + (−0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)3-s + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)9-s + (0.258 − 0.965i)11-s + (0.707 + 0.707i)13-s + (0.707 − 0.707i)15-s + (−0.965 − 0.258i)17-s + (−0.258 − 0.965i)19-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s + (0.707 − 0.707i)27-s + (0.707 + 0.707i)29-s + (−0.5 − 0.866i)31-s + (0.866 + 0.5i)33-s + (−0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.906 + 0.422i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ -0.906 + 0.422i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08790759169 + 0.3965746032i\)
\(L(\frac12)\) \(\approx\) \(0.08790759169 + 0.3965746032i\)
\(L(1)\) \(\approx\) \(0.6718444502 + 0.1720022075i\)
\(L(1)\) \(\approx\) \(0.6718444502 + 0.1720022075i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (-0.965 - 0.258i)T \)
19 \( 1 + (-0.258 - 0.965i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.258 - 0.965i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 + (0.965 + 0.258i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.965 + 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.965 + 0.258i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−20.664633216134281353896476756050, −20.086075396093206319366064171067, −19.34424422197317065618206106989, −18.67745807175947121574179614642, −17.865151666127047901401418832045, −17.444742773708915993757700779155, −16.22769211616538215554532951732, −15.58783061676695800370519211200, −14.6123504688054209534209506567, −14.048411335454510892508158603622, −12.79801593533848238552228487016, −12.48990248159403218647090147081, −11.575985551306335375456462831209, −10.83666843169469796629188416998, −10.09646530920239938021098010191, −8.57777575427922546516151003490, −8.163731372089851349463500038226, −7.162932106460145883619469817732, −6.61320929786989682560084967845, −5.73482071630768368291064518357, −4.51349868902250769837292097567, −3.63348433188487315309409347407, −2.50829277045441817961263831211, −1.60666906959357161298794138218, −0.19003021141646004815033635181, 1.15818007069354049341793548937, 2.80292382973050775934174474591, 3.77243678219684694568849590235, 4.33860574988941675110509852029, 5.21461625381054960141177184259, 6.18389945426302449809876434859, 7.09288297724023656503638279755, 8.47026390091765559569658401103, 8.76532610068090096871120849324, 9.62336438674155055599465025864, 10.81407015263600154017326309751, 11.414312870619063131846085874859, 11.80297460540355780705126124571, 13.11778408084958515867243101145, 13.83133493666306959570601135882, 14.85167228586681587365842096617, 15.67406545121738757656837491920, 16.087923672965530793878925910633, 16.77715478875142456591851087396, 17.60205495122777752486601037460, 18.60356564852690904573856763615, 19.58166391144159570452397017048, 20.04521606475923183601946205979, 20.91847288878139721313428122727, 21.68459511843200252041781529117

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