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Properties

Label 1-2552-2552.1341-r1-0-0
Degree $1$
Conductor $2552$
Sign $0.226 + 0.974i$
Analytic cond. $274.250$
Root an. cond. $274.250$
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  + (−0.623 − 0.781i)3-s + (0.900 − 0.433i)5-s + (−0.623 − 0.781i)7-s + (−0.222 + 0.974i)9-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)15-s − 17-s + (0.623 − 0.781i)19-s + (−0.222 + 0.974i)21-s + (−0.900 − 0.433i)23-s + (0.623 − 0.781i)25-s + (0.900 − 0.433i)27-s + (−0.900 + 0.433i)31-s + (−0.900 − 0.433i)35-s + (0.222 − 0.974i)37-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)3-s + (0.900 − 0.433i)5-s + (−0.623 − 0.781i)7-s + (−0.222 + 0.974i)9-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)15-s − 17-s + (0.623 − 0.781i)19-s + (−0.222 + 0.974i)21-s + (−0.900 − 0.433i)23-s + (0.623 − 0.781i)25-s + (0.900 − 0.433i)27-s + (−0.900 + 0.433i)31-s + (−0.900 − 0.433i)35-s + (0.222 − 0.974i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2552 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2552 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2552\)    =    \(2^{3} \cdot 11 \cdot 29\)
Sign: $0.226 + 0.974i$
Analytic conductor: \(274.250\)
Root analytic conductor: \(274.250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2552} (1341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2552,\ (1:\ ),\ 0.226 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4374513856 - 0.3474694440i\)
\(L(\frac12)\) \(\approx\) \(-0.4374513856 - 0.3474694440i\)
\(L(1)\) \(\approx\) \(0.6324204407 - 0.4601828386i\)
\(L(1)\) \(\approx\) \(0.6324204407 - 0.4601828386i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.623 - 0.781i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
7 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 - T \)
19 \( 1 + (0.623 - 0.781i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (0.222 - 0.974i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 - T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 + (0.222 - 0.974i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + (0.222 - 0.974i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−19.92679714246606775184161821744, −18.805417961108913928908314059699, −18.33817063242767549773131726866, −17.68221544866962162091453240678, −16.82378538474110855006760721500, −16.35341181360689756934782049345, −15.5431258435234878183778121313, −14.932523733668719970189451611653, −14.147205624594949649078826416192, −13.41513512505729851987195812125, −12.51225487856689098621928409070, −11.74092501423630584239309646371, −11.19519671920661026697375331578, −10.22907745110694353806547823339, −9.61178813095118518494111220767, −9.28212468020250679090248893197, −8.30502281598766712973300518794, −6.924743881157261326746967259075, −6.40870023283921122914788403835, −5.73132854379916355621374361942, −5.08624031715456290477148009818, −4.0843096805616141483422831119, −3.26117127352213563525251750060, −2.3411338092366436907886204541, −1.47625553467927495648344286579, 0.13815823017164862091497878831, 0.63527541483825183282543278164, 1.72595040801752032864104081767, 2.4596623304168722540940993353, 3.523160502990998717180619372489, 4.72374202674809423758666269870, 5.33655404912566425692759737666, 6.14418738959669450830711803976, 6.80475868184647027982523748519, 7.44190115450936796737629237261, 8.40373989964673141427140792668, 9.21444897152252609261570178952, 10.19377459821818872504285976472, 10.56210673360462613467677896358, 11.50685026294680543348910297079, 12.42522943799687711831285754775, 12.976998823006189768349565773000, 13.54072796564745601893843722700, 13.99908877822702437216230218522, 15.16624348031035968181955982583, 16.131963754981436496194765254569, 16.66637205127821434982019885267, 17.316618084631068803775780088238, 18.06930074874659199539425677590, 18.23816977534648313052581958729

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