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Properties

Label 1-5600-5600.117-r0-0-0
Degree $1$
Conductor $5600$
Sign $0.823 + 0.567i$
Analytic cond. $26.0062$
Root an. cond. $26.0062$
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  + (−0.0523 + 0.998i)3-s + (−0.994 − 0.104i)9-s + (0.629 − 0.777i)11-s + (0.156 + 0.987i)13-s + (0.207 − 0.978i)17-s + (0.0523 + 0.998i)19-s + (−0.913 − 0.406i)23-s + (0.156 − 0.987i)27-s + (0.453 + 0.891i)29-s + (0.978 + 0.207i)31-s + (0.743 + 0.669i)33-s + (0.629 + 0.777i)37-s + (−0.994 + 0.104i)39-s + (0.587 − 0.809i)41-s + (0.707 − 0.707i)43-s + ⋯
L(s)  = 1  + (−0.0523 + 0.998i)3-s + (−0.994 − 0.104i)9-s + (0.629 − 0.777i)11-s + (0.156 + 0.987i)13-s + (0.207 − 0.978i)17-s + (0.0523 + 0.998i)19-s + (−0.913 − 0.406i)23-s + (0.156 − 0.987i)27-s + (0.453 + 0.891i)29-s + (0.978 + 0.207i)31-s + (0.743 + 0.669i)33-s + (0.629 + 0.777i)37-s + (−0.994 + 0.104i)39-s + (0.587 − 0.809i)41-s + (0.707 − 0.707i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5600\)    =    \(2^{5} \cdot 5^{2} \cdot 7\)
Sign: $0.823 + 0.567i$
Analytic conductor: \(26.0062\)
Root analytic conductor: \(26.0062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5600} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5600,\ (0:\ ),\ 0.823 + 0.567i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.655852559 + 0.5155462544i\)
\(L(\frac12)\) \(\approx\) \(1.655852559 + 0.5155462544i\)
\(L(1)\) \(\approx\) \(1.044642820 + 0.2912772368i\)
\(L(1)\) \(\approx\) \(1.044642820 + 0.2912772368i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.0523 + 0.998i)T \)
11 \( 1 + (0.629 - 0.777i)T \)
13 \( 1 + (0.156 + 0.987i)T \)
17 \( 1 + (0.207 - 0.978i)T \)
19 \( 1 + (0.0523 + 0.998i)T \)
23 \( 1 + (-0.913 - 0.406i)T \)
29 \( 1 + (0.453 + 0.891i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.629 + 0.777i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.207 - 0.978i)T \)
53 \( 1 + (0.998 + 0.0523i)T \)
59 \( 1 + (-0.933 - 0.358i)T \)
61 \( 1 + (-0.933 + 0.358i)T \)
67 \( 1 + (-0.544 - 0.838i)T \)
71 \( 1 + (0.951 + 0.309i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (0.453 - 0.891i)T \)
89 \( 1 + (-0.406 + 0.913i)T \)
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

−17.77643196865040550312075823074, −17.36086727440193423400575158122, −16.68378084345334357298096082733, −15.646282792313897008834073719672, −15.1623294118466667793134527213, −14.36938575535719820274384462640, −13.80333835156554731817753625940, −12.93456230368054438896877134152, −12.67386773329900799040735963967, −11.846302098000757874948084034, −11.31338651668094099703556599266, −10.48263761426548191888417609225, −9.717831946384937921357729331500, −8.99719322528474654856796817241, −8.05398758014084581169937516005, −7.77682358249827081565508929017, −6.94025153908382028673473794016, −6.10137165165734378743244081166, −5.844743178576212933310872117094, −4.68155342343043133347607379483, −4.01797365249736456523565678875, −2.934962180112498723365270107396, −2.37522965339894781603949636855, −1.43140006954977651716307401665, −0.75983395648412706837813788750, 0.62580610879675514037439535979, 1.711405525111031747414529623291, 2.73789782697842578431316193614, 3.447865489170707899192876661880, 4.14365851727116853209560544281, 4.71531527159457937289288069428, 5.63637236553329474447907554199, 6.191594287002154327344931995662, 6.94228223028525831007822325450, 7.97262640190560071203734168384, 8.67034958298904205071463712873, 9.18785806384901858624885092271, 9.892801850516419075964405077556, 10.52194231363942816200443998590, 11.21734562552308692129714832441, 11.973895344335735592304414328726, 12.21118434928285557173303906644, 13.751746975164122925787918020855, 13.930532417548428731978647966833, 14.52948096741150686337486899349, 15.36909493483004332046754985608, 16.078499036132125942856652290390, 16.56535337003938575935053423697, 16.89446891056284869082437695441, 17.91457175737987404382659811761

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