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Properties

Label 1-1400-1400.437-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.103 + 0.994i$
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  + (0.406 − 0.913i)3-s + (−0.669 − 0.743i)9-s + (−0.669 + 0.743i)11-s + (−0.951 − 0.309i)13-s + (0.994 + 0.104i)17-s + (−0.913 + 0.406i)19-s + (−0.207 + 0.978i)23-s + (−0.951 + 0.309i)27-s + (−0.809 + 0.587i)29-s + (0.104 − 0.994i)31-s + (0.406 + 0.913i)33-s + (0.743 − 0.669i)37-s + (−0.669 + 0.743i)39-s + (−0.309 + 0.951i)41-s − i·43-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)3-s + (−0.669 − 0.743i)9-s + (−0.669 + 0.743i)11-s + (−0.951 − 0.309i)13-s + (0.994 + 0.104i)17-s + (−0.913 + 0.406i)19-s + (−0.207 + 0.978i)23-s + (−0.951 + 0.309i)27-s + (−0.809 + 0.587i)29-s + (0.104 − 0.994i)31-s + (0.406 + 0.913i)33-s + (0.743 − 0.669i)37-s + (−0.669 + 0.743i)39-s + (−0.309 + 0.951i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.103 + 0.994i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.103 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4856533361 + 0.4377622745i\)
\(L(\frac12)\) \(\approx\) \(0.4856533361 + 0.4377622745i\)
\(L(1)\) \(\approx\) \(0.8935679745 - 0.1341872052i\)
\(L(1)\) \(\approx\) \(0.8935679745 - 0.1341872052i\)

Euler product

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

−20.84001073526831035201573386957, −19.89686942684390264485758678084, −19.16769965450273046338764363949, −18.587989803325348336268129783014, −17.370650250086590296132459637696, −16.68319163628864057629694142160, −16.1559192569292551742666556669, −15.220679212510578211817890291117, −14.62696700349382488595662795658, −13.92913573040252438264352643160, −13.07291099805781533964306352054, −12.120037587500794671890304556975, −11.22720733127036504193820703521, −10.40217024071420565116920046848, −9.89829847770793383214409001912, −8.906047534929690116767268758515, −8.27075717334865903719786039315, −7.441933276515475300750361852429, −6.29107375590654818446826818171, −5.29902438292880743486974267792, −4.6871422764909965006872811654, −3.68356884815433456127927919450, −2.8502125913044841715301037772, −2.03846438836521756185028765872, −0.22404846916521295746334189930, 1.31182683743935232570727077094, 2.21406456122821320773985426698, 2.9881188061336644775670253714, 4.05272271545497036220739680175, 5.21925312206809125078066748215, 5.977698863251279596472257645784, 6.99620666728581862961156740559, 7.771428212741989350069641833301, 8.11404099475208781546666658319, 9.461983674585166744097634876966, 9.89749570178551123210966326641, 11.07979476889524641437376843101, 11.934001019154833056175695171955, 12.855611543283296994169682561589, 12.95998314633928893668754203911, 14.24760555012034116634700686332, 14.75735853061672338246945066259, 15.42570769982021643946442257158, 16.66638298002965992663900778109, 17.26446934647358055199020862261, 18.111804170292204638667130882796, 18.66086340368915257476410286938, 19.5230740887463410325028999432, 20.04141219646963136552415857726, 20.889426302146522574338787962979

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