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Properties

Label 1-1400-1400.1259-r0-0-0
Degree $1$
Conductor $1400$
Sign $-0.728 + 0.684i$
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  + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (0.809 + 0.587i)19-s + (0.309 + 0.951i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.309 − 0.951i)37-s + (−0.309 − 0.951i)39-s + (−0.309 + 0.951i)41-s − 43-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (0.809 + 0.587i)19-s + (0.309 + 0.951i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.309 − 0.951i)37-s + (−0.309 − 0.951i)39-s + (−0.309 + 0.951i)41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.728 + 0.684i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.728 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2977571462 + 0.7520487048i\)
\(L(\frac12)\) \(\approx\) \(0.2977571462 + 0.7520487048i\)
\(L(1)\) \(\approx\) \(0.7221326428 + 0.2848084530i\)
\(L(1)\) \(\approx\) \(0.7221326428 + 0.2848084530i\)

Euler product

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

−20.29390462296056126111371854971, −19.74796010755318946298843395153, −18.869488665124927950295610367151, −18.22225312663555221942211487673, −17.47867990837437368902450336718, −16.88881028330962561046353982577, −16.04721854418320410979514396578, −15.36712249263788001374951083125, −14.24285257347146488830091186045, −13.548958981450446141005987003238, −12.723688680467698090858529428407, −12.186515825917688231649869199740, −11.06007503949513122688428066667, −10.83936824358043157660242348495, −9.75782820326849368874081970137, −8.60660299288891289535537854827, −8.01070520881479578264696279517, −6.88164620538490012037529109449, −6.417575898804318473661725013007, −5.37589407224220674217814905370, −4.82446781784669452384276413783, −3.51053705957424648359127597708, −2.559169628854838907872072921683, −1.37062261957208526915186251596, −0.38815363708615925878766840913, 1.21647354476384935953513521472, 2.30706932129685775159708270826, 3.628245026863965205592461305974, 4.4038423295302707176007987372, 5.07776171030792914860048967882, 6.018244214105354419777466910562, 6.90468748771934280398847527177, 7.515242880953383312528039051, 8.94777995252210802366149736025, 9.568105025608355694451460684216, 10.13174168699869502841236833466, 11.28426731703924517484672053368, 11.7004972843655607360898004518, 12.44064095659550703110419111856, 13.438520473646062802752278978941, 14.35844951454028312242777591275, 15.15441926059234374406327576180, 15.8247803797557388230982540666, 16.596510230706871359265947213774, 17.239379854190582181960699905692, 17.987215754896765635759899877426, 18.599223939332900244785417676968, 19.77764236468783732848635347846, 20.32653444961343834387076714365, 21.25944534893185014560591759387

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