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Properties

Label 1-1400-1400.261-r0-0-0
Degree $1$
Conductor $1400$
Sign $-0.729 + 0.684i$
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  + (−0.913 + 0.406i)3-s + (0.669 − 0.743i)9-s + (−0.669 − 0.743i)11-s + (−0.309 − 0.951i)13-s + (−0.104 − 0.994i)17-s + (−0.913 − 0.406i)19-s + (−0.978 + 0.207i)23-s + (−0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.104 − 0.994i)31-s + (0.913 + 0.406i)33-s + (−0.669 + 0.743i)37-s + (0.669 + 0.743i)39-s + (0.309 + 0.951i)41-s − 43-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)3-s + (0.669 − 0.743i)9-s + (−0.669 − 0.743i)11-s + (−0.309 − 0.951i)13-s + (−0.104 − 0.994i)17-s + (−0.913 − 0.406i)19-s + (−0.978 + 0.207i)23-s + (−0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.104 − 0.994i)31-s + (0.913 + 0.406i)33-s + (−0.669 + 0.743i)37-s + (0.669 + 0.743i)39-s + (0.309 + 0.951i)41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.729 + 0.684i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.729 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06552382286 + 0.1656257339i\)
\(L(\frac12)\) \(\approx\) \(0.06552382286 + 0.1656257339i\)
\(L(1)\) \(\approx\) \(0.6131424590 + 0.003597146569i\)
\(L(1)\) \(\approx\) \(0.6131424590 + 0.003597146569i\)

Euler product

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

−20.62725792823519232922709294079, −19.41508470590093280942698722964, −19.116869625131851678737090441928, −18.024877952679216012569779107672, −17.61760805053166258880980144848, −16.78542221953065001051187569328, −16.11125224149659840804539023767, −15.31002377359820357903812857335, −14.37587562465456816970754490282, −13.53158844866666387301521292433, −12.54504442799953492144383833860, −12.27111633025659927165010616317, −11.31038618390875857065729688047, −10.38002017949780618369717847716, −10.018927461802807017749289142981, −8.67382164748661326538776363121, −7.9014253181159909869458951834, −6.9321677421022459237855622007, −6.38803367740296286097189141996, −5.42793585073109553743100268652, −4.60316913667028181331727289735, −3.83628995623129072781897110284, −2.17866531126621600162978023167, −1.749341137456421164337213792139, −0.08822565766470317247322695731, 0.98751124885211763123778248147, 2.49532412506332253139466739644, 3.38926326544005780972943672745, 4.51459161407553312502809232952, 5.1926619372951194010433656990, 5.96613060896678704659220980018, 6.755926143410593515438599676157, 7.78130119075309673635971123194, 8.59495052277882372439993968966, 9.731944167203909193481915915293, 10.2745455827653085925696869968, 11.111741007553947981547439813085, 11.69595154984957317943350725361, 12.66661585929875996392417760618, 13.26405184414765684712805314738, 14.282710366222728739844323849, 15.27369895956143341740438551628, 15.84009511879539156214407448388, 16.48993968448531419935742876755, 17.34589313854122461586954225539, 17.9869911261996331968743799209, 18.61824430487308530416203270715, 19.58408695650262938528466421445, 20.519376151216842555408201637601, 21.10824549956326620936131744976

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