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Properties

Label 1-71e2-5041.1634-r0-0-0
Degree $1$
Conductor $5041$
Sign $-0.906 - 0.423i$
Analytic cond. $23.4103$
Root an. cond. $23.4103$
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  + (0.699 − 0.714i)2-s + (0.984 − 0.176i)3-s + (−0.0221 − 0.999i)4-s + (0.964 − 0.262i)5-s + (0.562 − 0.826i)6-s + (−0.787 − 0.616i)7-s + (−0.730 − 0.683i)8-s + (0.937 − 0.346i)9-s + (0.487 − 0.873i)10-s + 11-s + (−0.197 − 0.980i)12-s + (0.699 + 0.714i)13-s + (−0.991 + 0.132i)14-s + (0.903 − 0.428i)15-s + (−0.999 + 0.0442i)16-s + (−0.283 − 0.958i)17-s + ⋯
L(s)  = 1  + (0.699 − 0.714i)2-s + (0.984 − 0.176i)3-s + (−0.0221 − 0.999i)4-s + (0.964 − 0.262i)5-s + (0.562 − 0.826i)6-s + (−0.787 − 0.616i)7-s + (−0.730 − 0.683i)8-s + (0.937 − 0.346i)9-s + (0.487 − 0.873i)10-s + 11-s + (−0.197 − 0.980i)12-s + (0.699 + 0.714i)13-s + (−0.991 + 0.132i)14-s + (0.903 − 0.428i)15-s + (−0.999 + 0.0442i)16-s + (−0.283 − 0.958i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5041\)    =    \(71^{2}\)
Sign: $-0.906 - 0.423i$
Analytic conductor: \(23.4103\)
Root analytic conductor: \(23.4103\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5041} (1634, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5041,\ (0:\ ),\ -0.906 - 0.423i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9627814147 - 4.337318348i\)
\(L(\frac12)\) \(\approx\) \(0.9627814147 - 4.337318348i\)
\(L(1)\) \(\approx\) \(1.711581590 - 1.602489250i\)
\(L(1)\) \(\approx\) \(1.711581590 - 1.602489250i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.699 - 0.714i)T \)
3 \( 1 + (0.984 - 0.176i)T \)
5 \( 1 + (0.964 - 0.262i)T \)
7 \( 1 + (-0.787 - 0.616i)T \)
11 \( 1 + T \)
13 \( 1 + (0.699 + 0.714i)T \)
17 \( 1 + (-0.283 - 0.958i)T \)
19 \( 1 + (-0.448 + 0.894i)T \)
23 \( 1 + (-0.730 - 0.683i)T \)
29 \( 1 + (0.937 + 0.346i)T \)
31 \( 1 + (-0.921 - 0.387i)T \)
37 \( 1 + (-0.730 - 0.683i)T \)
41 \( 1 + (-0.952 + 0.304i)T \)
43 \( 1 + (0.0663 - 0.997i)T \)
47 \( 1 + (0.814 - 0.580i)T \)
53 \( 1 + (-0.921 - 0.387i)T \)
59 \( 1 + (-0.525 + 0.850i)T \)
61 \( 1 + (0.154 - 0.988i)T \)
67 \( 1 + (-0.787 - 0.616i)T \)
73 \( 1 + (0.562 + 0.826i)T \)
79 \( 1 + (-0.367 - 0.930i)T \)
83 \( 1 + (0.814 + 0.580i)T \)
89 \( 1 + (0.903 + 0.428i)T \)
97 \( 1 + (0.814 + 0.580i)T \)
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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

−18.16805140503616957683039872788, −17.568225922645695129112587423413, −16.95415426675308481665276107980, −16.031598881370696351534589614193, −15.51260759917384835264433131774, −15.01908844364008087356875273750, −14.30701612090703762576035187938, −13.73555731635263038439632170216, −13.15223130604838247713525057545, −12.71479769101787109208207337381, −11.89558829538253066784586080129, −10.81504687359161902321796135809, −10.08618238237090944217149518338, −9.20385521999300209313774722319, −8.86581322301526816530946251719, −8.19636082077008383979584515949, −7.23982412223330187626101509022, −6.3736012370143848131816948514, −6.21611839025399308588693448772, −5.25569387032903027670362277634, −4.37305473965231588941229171186, −3.51042412455569916281469744960, −3.1011366183546689725623306606, −2.26131964107241716476365889744, −1.486921284259283574760991017054, 0.70188323915690657574257878114, 1.64780529827561567761062085309, 2.0485850418632649725948986866, 3.00404514754887303748201663896, 3.76757648485014701247206087225, 4.175110989503928999088456664849, 5.102478765267322951265907510034, 6.2699030029405981826037315614, 6.48279058687990958775297335633, 7.25628246138857850501223622695, 8.60559532366905514285615800340, 9.087504464996697996875301792522, 9.64723549420880589288686767475, 10.27097573402200550354865563631, 10.880263175635726734073547702584, 12.13797709116014473517534675828, 12.35942868152642766790013876865, 13.35706026794191892700655302321, 13.667327300399008386691254554093, 14.22521003244825298935726079120, 14.60876351786754700227454284728, 15.73266688642671331978010538087, 16.28260620186664524086479180837, 17.0061362814041043351322908999, 18.08127299187019923064877743052

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