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Properties

Label 1-5200-5200.4163-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.0929 + 0.995i$
Analytic cond. $24.1486$
Root an. cond. $24.1486$
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  + (−0.104 − 0.994i)3-s + (−0.866 + 0.5i)7-s + (−0.978 + 0.207i)9-s + (0.207 − 0.978i)11-s + (0.994 + 0.104i)17-s + (−0.994 − 0.104i)19-s + (0.587 + 0.809i)21-s + (0.207 − 0.978i)23-s + (0.309 + 0.951i)27-s + (−0.994 + 0.104i)29-s + (0.809 + 0.587i)31-s + (−0.994 − 0.104i)33-s + (−0.669 − 0.743i)37-s + (−0.669 − 0.743i)41-s + (0.5 + 0.866i)43-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)3-s + (−0.866 + 0.5i)7-s + (−0.978 + 0.207i)9-s + (0.207 − 0.978i)11-s + (0.994 + 0.104i)17-s + (−0.994 − 0.104i)19-s + (0.587 + 0.809i)21-s + (0.207 − 0.978i)23-s + (0.309 + 0.951i)27-s + (−0.994 + 0.104i)29-s + (0.809 + 0.587i)31-s + (−0.994 − 0.104i)33-s + (−0.669 − 0.743i)37-s + (−0.669 − 0.743i)41-s + (0.5 + 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0929 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0929 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.0929 + 0.995i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (4163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.0929 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1779267502 + 0.1620922079i\)
\(L(\frac12)\) \(\approx\) \(0.1779267502 + 0.1620922079i\)
\(L(1)\) \(\approx\) \(0.7275014590 - 0.2453519000i\)
\(L(1)\) \(\approx\) \(0.7275014590 - 0.2453519000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.104 - 0.994i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.207 - 0.978i)T \)
17 \( 1 + (0.994 + 0.104i)T \)
19 \( 1 + (-0.994 - 0.104i)T \)
23 \( 1 + (0.207 - 0.978i)T \)
29 \( 1 + (-0.994 + 0.104i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.669 - 0.743i)T \)
41 \( 1 + (-0.669 - 0.743i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.207 + 0.978i)T \)
61 \( 1 + (0.743 + 0.669i)T \)
67 \( 1 + (-0.913 + 0.406i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.978 - 0.207i)T \)
97 \( 1 + (0.406 - 0.913i)T \)
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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

−17.38981207714373373564808414270, −17.16843100611227125375706566138, −16.56833080189815357208500316732, −15.74440883541698354859941128755, −15.299379104402667945664505896651, −14.627384087815633065366464147297, −13.954262022867959587984932260816, −13.131686245847748572803909845344, −12.479146110682591164228879089498, −11.73945004239516578863659063614, −11.01862591192319627900821874406, −10.25793307848429947121199826438, −9.68250235154875199295136092326, −9.43422963466585380207272800482, −8.35856280333758804007693560283, −7.65331433887372397211879947837, −6.75637608829612210130635776621, −6.16422985971727744259468071384, −5.28257218433090159013870490047, −4.63304161503185050139968666410, −3.7894211572153898326598386792, −3.39481986915587230172771327708, −2.43342461517263778397525204268, −1.38922394792324044892727095366, −0.07555341791974761752801694200, 0.87494660965948780469609664427, 1.81551142140698625193103368656, 2.715161581416418917109044553542, 3.22889437924306711909158673821, 4.14050027991484254422065171255, 5.40891054413089540582058914947, 5.80076639455468837276223299252, 6.57009737371523937904203271242, 7.0208687723807813869356588540, 8.03129953821387154397434378686, 8.61002671050971662345179251073, 9.11937270094674037524598632689, 10.15920911674057167293822742133, 10.80712983178152611713648150678, 11.60690246679000298182753584638, 12.26124321496474881243542370043, 12.776862383271730952472336691892, 13.32704253427821068811399305273, 14.14058131446063468155906770407, 14.63358091525663243596852828011, 15.51415820786500226039265260210, 16.37428233541176478956587775146, 16.78441699136361572137642499075, 17.43018275899602307023730429631, 18.43122248697056589512050653670

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