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Properties

Label 1-1400-1400.397-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.0847 + 0.996i$
Analytic cond. $6.50157$
Root an. cond. $6.50157$
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  + (−0.207 + 0.978i)3-s + (−0.913 − 0.406i)9-s + (−0.913 + 0.406i)11-s + (0.587 + 0.809i)13-s + (0.743 − 0.669i)17-s + (0.978 − 0.207i)19-s + (0.994 + 0.104i)23-s + (0.587 − 0.809i)27-s + (0.309 + 0.951i)29-s + (−0.669 − 0.743i)31-s + (−0.207 − 0.978i)33-s + (0.406 − 0.913i)37-s + (−0.913 + 0.406i)39-s + (0.809 − 0.587i)41-s − i·43-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)3-s + (−0.913 − 0.406i)9-s + (−0.913 + 0.406i)11-s + (0.587 + 0.809i)13-s + (0.743 − 0.669i)17-s + (0.978 − 0.207i)19-s + (0.994 + 0.104i)23-s + (0.587 − 0.809i)27-s + (0.309 + 0.951i)29-s + (−0.669 − 0.743i)31-s + (−0.207 − 0.978i)33-s + (0.406 − 0.913i)37-s + (−0.913 + 0.406i)39-s + (0.809 − 0.587i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0847 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0847 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.0847 + 0.996i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.0847 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.002633634 + 0.9209639646i\)
\(L(\frac12)\) \(\approx\) \(1.002633634 + 0.9209639646i\)
\(L(1)\) \(\approx\) \(0.9379400426 + 0.3746999436i\)
\(L(1)\) \(\approx\) \(0.9379400426 + 0.3746999436i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.207 + 0.978i)T \)
11 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (0.743 - 0.669i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (0.994 + 0.104i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.406 - 0.913i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.743 - 0.669i)T \)
53 \( 1 + (0.207 - 0.978i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.743 - 0.669i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.406 + 0.913i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−20.55735596640784906021392334191, −19.80180912973291027694771894988, −18.87463131027382211060678370286, −18.484742561272418117557036119262, −17.70040548224918447209017969662, −16.97944394519251116133566948879, −16.133862705903401508404906735747, −15.34070540174985759735357318398, −14.364247667308588575589304671846, −13.60332076831276988730961307250, −12.93173787085820228151492511353, −12.36774061059792580553329961432, −11.35269713557483484189234842126, −10.755731662607342891884752822668, −9.85423858498356073410523509334, −8.64761294093116623638746308877, −7.97296327537867151213050898220, −7.42364593267837101486579458499, −6.29998212924351640749554189793, −5.667791061444486035859550582339, −4.930113729227086246749245721783, −3.37870059853824831605176825128, −2.82303156468550801401040686156, −1.56673874473596240191611005427, −0.68901420132803350622017361575, 0.98256505953206930481034231152, 2.460128985393729845518995615372, 3.30338706492736726700079873900, 4.19127877597005536899111433967, 5.15433483730212080713862751757, 5.59367152154587185904597626934, 6.83397326624138244199198481748, 7.640051200773790480186662150, 8.722453084132770538978864606393, 9.43834597324899294144093993440, 10.05588669407976111680399488839, 11.047488477682706865877604541330, 11.47038127400771881072859409231, 12.497789244459931844132468674669, 13.39565554657806667822856617400, 14.3150776469090963463163406074, 14.89145868802459406279927281700, 15.98540163268083578786624654539, 16.15407463762994363693188105812, 17.04440289910962192376260683488, 18.05681920376539746690126997858, 18.49559404882613065157452754073, 19.6611622154802412072481634640, 20.369674346458348839642177061311, 21.227289548742968183652426511075

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