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Properties

Label 1-4235-4235.1059-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.200 + 0.979i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (0.595 + 0.803i)2-s + (0.104 + 0.994i)3-s + (−0.290 + 0.956i)4-s + (−0.736 + 0.676i)6-s + (−0.941 + 0.336i)8-s + (−0.978 + 0.207i)9-s + (−0.981 − 0.189i)12-s + (−0.516 − 0.856i)13-s + (−0.830 − 0.556i)16-s + (0.935 − 0.353i)17-s + (−0.749 − 0.662i)18-s + (0.548 + 0.836i)19-s + (0.786 − 0.618i)23-s + (−0.432 − 0.901i)24-s + (0.380 − 0.924i)26-s + (−0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (0.595 + 0.803i)2-s + (0.104 + 0.994i)3-s + (−0.290 + 0.956i)4-s + (−0.736 + 0.676i)6-s + (−0.941 + 0.336i)8-s + (−0.978 + 0.207i)9-s + (−0.981 − 0.189i)12-s + (−0.516 − 0.856i)13-s + (−0.830 − 0.556i)16-s + (0.935 − 0.353i)17-s + (−0.749 − 0.662i)18-s + (0.548 + 0.836i)19-s + (0.786 − 0.618i)23-s + (−0.432 − 0.901i)24-s + (0.380 − 0.924i)26-s + (−0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.200 + 0.979i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.200 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.556438003 + 1.270702641i\)
\(L(\frac12)\) \(\approx\) \(1.556438003 + 1.270702641i\)
\(L(1)\) \(\approx\) \(1.024637462 + 0.8532450813i\)
\(L(1)\) \(\approx\) \(1.024637462 + 0.8532450813i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.595 + 0.803i)T \)
3 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (-0.516 - 0.856i)T \)
17 \( 1 + (0.935 - 0.353i)T \)
19 \( 1 + (0.548 + 0.836i)T \)
23 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (-0.362 - 0.931i)T \)
31 \( 1 + (0.123 - 0.992i)T \)
37 \( 1 + (-0.820 + 0.572i)T \)
41 \( 1 + (-0.870 - 0.491i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (-0.749 + 0.662i)T \)
53 \( 1 + (0.830 - 0.556i)T \)
59 \( 1 + (0.00951 - 0.999i)T \)
61 \( 1 + (-0.398 - 0.917i)T \)
67 \( 1 + (0.995 + 0.0950i)T \)
71 \( 1 + (-0.254 + 0.967i)T \)
73 \( 1 + (0.532 + 0.846i)T \)
79 \( 1 + (0.879 - 0.475i)T \)
83 \( 1 + (0.985 - 0.170i)T \)
89 \( 1 + (-0.888 - 0.458i)T \)
97 \( 1 + (0.564 - 0.825i)T \)
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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

−18.39765660686334670925268298666, −17.80505892784389358273708424975, −17.017233568649744121265283510199, −16.23768555726717966070982195552, −15.13586498500376216309777826667, −14.67086886798767379515618737046, −13.84303388565620494693659042495, −13.56783208184010988829919017303, −12.6304296483401117820990850567, −12.18836936125227275282126121030, −11.60175862267875923752123296617, −10.90165099396959473597704008663, −10.124490790913673339792133926576, −9.11261992104980777217439534287, −8.852003778355728601393242777055, −7.59395795525452397428060434483, −7.00447994464970504922301990259, −6.32681490517651571459859083924, −5.31731657019621982288784144529, −4.99543732040951041215759793503, −3.71641169631064539471703265804, −3.15482314274747128241425838391, −2.326674552094056728726368774405, −1.55407518210908287190366454949, −0.8913530726962788961563652973, 0.53358119156962466956708330293, 2.262602514908529835601404191242, 3.1390418666738156203079854669, 3.59375185469089932706230836993, 4.497406304815016385262965410361, 5.20653115719196151101032486680, 5.63969632366485491666647258068, 6.47209972319196342327381606024, 7.51644832528142856218068121344, 8.01566293276538387562119853080, 8.70412665453577608600160889048, 9.69690942080422830439857761949, 9.99046561642115856052566273089, 11.06936315964688351603898640504, 11.777837297856560007681883455579, 12.47096095096399061037641252616, 13.21026514930037793662858785954, 14.1207029979331765266756676041, 14.46986065096218911107195697885, 15.2450040548893666559321096921, 15.65066747151472167916337676637, 16.43376438819009065620468111573, 17.015400057709659051773597567975, 17.410746859011632774858507855836, 18.44374580251737874831909317737

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