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Properties

Label 1-1400-1400.979-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.535 - 0.844i$
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  + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.809 − 0.587i)23-s + (−0.809 − 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.309 − 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.809 + 0.587i)37-s + (0.809 + 0.587i)39-s + (0.809 − 0.587i)41-s − 43-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.809 − 0.587i)23-s + (−0.809 − 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.309 − 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.809 + 0.587i)37-s + (0.809 + 0.587i)39-s + (0.809 − 0.587i)41-s − 43-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9220998257 - 0.5069286690i\)
\(L(\frac12)\) \(\approx\) \(0.9220998257 - 0.5069286690i\)
\(L(1)\) \(\approx\) \(0.9843611307 + 0.09495567906i\)
\(L(1)\) \(\approx\) \(0.9843611307 + 0.09495567906i\)

Euler product

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

−20.97537294268413925563753838653, −19.88289721805728774149887389896, −19.52915127089093264043105774607, −18.59111573255270003495496586775, −17.955503152898166630450300359340, −17.441731840170643337956374631910, −16.35664380794879981174058834938, −15.57183716205033937587496292199, −14.710081543180149072814070905630, −13.993875147507519756231099517160, −13.15141852996028462069349786335, −12.69621218297423144472765097172, −11.799217395720199766594715717729, −10.98692079425548935664082589138, −10.109867702281873226082516313255, −9.013195077518514083970809886176, −8.40878774762996442722200695173, −7.56208294847084184970698416117, −6.82201786094580469129136697836, −6.04126675416308866360113693762, −5.11332809445920805894085728713, −3.94599777216310598585390480098, −3.02234168208613711140208270794, −2.0138592016419260282269561726, −1.29069183515252285985596646499, 0.3793326036531216268687319933, 2.048348073227022049068746300533, 3.016144161893860376760439980426, 3.722160386509534136595802017157, 4.64967685930236680558363783348, 5.59549770510112439663864969009, 6.14540370942424902155624837376, 7.66238745635375089408381349031, 8.215128165872241547945332902271, 8.96734288880469271683753882039, 10.04516546332036832171996828535, 10.39920899387858475108017808134, 11.32206101781172727123798037269, 12.0867245755580616516088668239, 13.33700667646860654109114334038, 13.75730179250562611109679728614, 14.70725988310488302741333099474, 15.43786568256616216526046504438, 16.12840883641487094007559303662, 16.614105029394921872098117659066, 17.6192856590153760145124284455, 18.57854628605329077788306168001, 19.05834417892700601962273332100, 20.32590352938588751755573252202, 20.62944086350305919106354602297

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