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Properties

Label 1-1400-1400.853-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.844 + 0.535i$
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  + (−0.951 + 0.309i)3-s + (0.809 − 0.587i)9-s + (0.809 + 0.587i)11-s + (−0.587 − 0.809i)13-s + (0.951 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (0.587 − 0.809i)23-s + (−0.587 + 0.809i)27-s + (0.309 + 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.587 − 0.809i)37-s + (0.809 + 0.587i)39-s + (0.809 − 0.587i)41-s − i·43-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)3-s + (0.809 − 0.587i)9-s + (0.809 + 0.587i)11-s + (−0.587 − 0.809i)13-s + (0.951 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (0.587 − 0.809i)23-s + (−0.587 + 0.809i)27-s + (0.309 + 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.587 − 0.809i)37-s + (0.809 + 0.587i)39-s + (0.809 − 0.587i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.844 + 0.535i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.844 + 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.093069759 + 0.3175661213i\)
\(L(\frac12)\) \(\approx\) \(1.093069759 + 0.3175661213i\)
\(L(1)\) \(\approx\) \(0.8595497209 + 0.1097881126i\)
\(L(1)\) \(\approx\) \(0.8595497209 + 0.1097881126i\)

Euler product

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

−21.07048835307896116449635825007, −19.55484469926925070183516240474, −19.32818259783840341125098265150, −18.44249575575454296936594566517, −17.6226532541211938436284352443, −16.816013989064876607810894524821, −16.573252667511982782236092329426, −15.49133048765855798028825545855, −14.658777310003659640599792013233, −13.69727054771210661566491971721, −13.10977952605070296948759743327, −11.99679198482486678060427049509, −11.629016144002987104456717373576, −10.93294908611191148764141303467, −9.82196583690047499710085626324, −9.26732389500345808607668937092, −8.06659857547289456555793424680, −7.200020722118828785786958213795, −6.51690345339153737335104686201, −5.73167005652174782483187662473, −4.83952454507661257076530977821, −4.06007441380048104557557208703, −2.84449036476681089849373551981, −1.66772114137797101587448616826, −0.71438378885426752368954331268, 0.85393212484329976258424732308, 1.87523079622243170351192274688, 3.3049710617652834660409473200, 4.087470942048589069841596637351, 5.091338939497620796037637554776, 5.65644782076122410367220670422, 6.69402698580425139282768508448, 7.29088071470357713412825987518, 8.41503867521044637619125797990, 9.384546241101395615908055031170, 10.264894369320749318895252404013, 10.62941479108255009491738213641, 11.74348941037240666956084844338, 12.542955861326721838519932170564, 12.68476302279130183949177338912, 14.38809868230009429314411019321, 14.64981337543997142986602936059, 15.69343430286823474864333715513, 16.44034102702196197433418661396, 17.12546755984917462477392974969, 17.6584384503632171464127613410, 18.4990711865761479634183823877, 19.30722063112235837766364336305, 20.19303407271438691883306474632, 20.97835397207462386922276377613

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