Properties

Label 1-4235-4235.2979-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.996 + 0.0845i$
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  + (0.991 − 0.132i)2-s + (0.978 − 0.207i)3-s + (0.964 − 0.263i)4-s + (0.941 − 0.336i)6-s + (0.921 − 0.389i)8-s + (0.913 − 0.406i)9-s + (0.888 − 0.458i)12-s + (0.362 + 0.931i)13-s + (0.861 − 0.508i)16-s + (−0.272 + 0.962i)17-s + (0.851 − 0.524i)18-s + (−0.830 + 0.556i)19-s + (0.995 + 0.0950i)23-s + (0.820 − 0.572i)24-s + (0.483 + 0.875i)26-s + (0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (0.991 − 0.132i)2-s + (0.978 − 0.207i)3-s + (0.964 − 0.263i)4-s + (0.941 − 0.336i)6-s + (0.921 − 0.389i)8-s + (0.913 − 0.406i)9-s + (0.888 − 0.458i)12-s + (0.362 + 0.931i)13-s + (0.861 − 0.508i)16-s + (−0.272 + 0.962i)17-s + (0.851 − 0.524i)18-s + (−0.830 + 0.556i)19-s + (0.995 + 0.0950i)23-s + (0.820 − 0.572i)24-s + (0.483 + 0.875i)26-s + (0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.996 + 0.0845i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.996 + 0.0845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.684607220 + 0.2406513235i\)
\(L(\frac12)\) \(\approx\) \(5.684607220 + 0.2406513235i\)
\(L(1)\) \(\approx\) \(2.867713967 - 0.1551135257i\)
\(L(1)\) \(\approx\) \(2.867713967 - 0.1551135257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.991 - 0.132i)T \)
3 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (0.362 + 0.931i)T \)
17 \( 1 + (-0.272 + 0.962i)T \)
19 \( 1 + (-0.830 + 0.556i)T \)
23 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (-0.985 + 0.170i)T \)
31 \( 1 + (0.449 + 0.893i)T \)
37 \( 1 + (0.935 - 0.353i)T \)
41 \( 1 + (-0.564 + 0.825i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (0.851 + 0.524i)T \)
53 \( 1 + (-0.861 - 0.508i)T \)
59 \( 1 + (-0.432 + 0.901i)T \)
61 \( 1 + (0.380 + 0.924i)T \)
67 \( 1 + (-0.235 - 0.971i)T \)
71 \( 1 + (-0.466 - 0.884i)T \)
73 \( 1 + (-0.953 - 0.299i)T \)
79 \( 1 + (-0.290 + 0.956i)T \)
83 \( 1 + (-0.198 - 0.980i)T \)
89 \( 1 + (0.928 + 0.371i)T \)
97 \( 1 + (-0.0855 - 0.996i)T \)
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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

−18.730121644352465832912876602128, −17.46754203567986576080246470207, −16.919252323264033966606060926159, −15.912412368874064419695863407954, −15.51623173193368165930362825546, −14.9561146913158050315527464469, −14.35067380149718118080332993161, −13.48042116433375143660887683876, −13.18849756057865846596456814615, −12.5594922555111616200188208558, −11.55251442864940683364646070820, −10.894053259463343397236528315070, −10.2468144449683538487522456600, −9.29880709578686702439321173452, −8.622424574887205806527464065461, −7.78453437903773266111288031984, −7.25558110041490786249423092946, −6.48007044296677166082247167379, −5.55378563177753447861390272323, −4.82373564705643223219352347991, −4.130033058971941173768908957367, −3.400916639298569800271485621314, −2.649972729613280580540505272487, −2.1574800834995069527294827997, −0.93933474443705697959990573617, 1.3545737678435699960587942465, 1.77847092888123504445576484978, 2.72116513890172272198498318083, 3.38858720603610432579293009603, 4.221839763806163405302368620488, 4.5826994338104747294820792462, 5.8610639961723299098598032167, 6.42266001419059226600067637149, 7.14697334689425271325464558002, 7.862351843404220504746590314, 8.688321694379791224221925158604, 9.34611232244735642182384448193, 10.27929923777248371264029401596, 10.92537184898789540122243127666, 11.66306947399117388330487139663, 12.65932801470722469782121133659, 12.88085007850172787066111321899, 13.6809295981306889679142016453, 14.278617359230326915012190592288, 14.94385816815578294360385341303, 15.264057167811054746112203576664, 16.27394836114610251892995938129, 16.72679103784850596303804854059, 17.76318634129176291037339720095, 18.74311296341228203063369458808

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