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Properties

Label 1-52-52.35-r1-0-0
Degree $1$
Conductor $52$
Sign $0.711 + 0.702i$
Analytic cond. $5.58817$
Root an. cond. $5.58817$
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  + (0.5 + 0.866i)3-s + 5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + 21-s + (0.5 + 0.866i)23-s + 25-s − 27-s + (−0.5 − 0.866i)29-s − 31-s + (−0.5 + 0.866i)33-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + 5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + 21-s + (0.5 + 0.866i)23-s + 25-s − 27-s + (−0.5 − 0.866i)29-s − 31-s + (−0.5 + 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(5.58817\)
Root analytic conductor: \(5.58817\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 52,\ (1:\ ),\ 0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.982770984 + 0.8138532600i\)
\(L(\frac12)\) \(\approx\) \(1.982770984 + 0.8138532600i\)
\(L(1)\) \(\approx\) \(1.460222993 + 0.3812477201i\)
\(L(1)\) \(\approx\) \(1.460222993 + 0.3812477201i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−32.990703130516555897941466339846, −31.730202550868152344861877412811, −30.80804044061694231996438982436, −29.55025386832094499887010616903, −28.902220984479481328630339790239, −27.28069928844560711590175100291, −25.89532386794198401258343807136, −24.78142970527927812691140697738, −24.39172655741410356227758086134, −22.54993655599350479127027143738, −21.34242550617004668704884155074, −20.23039383070662110173127043504, −18.648597134081762969207911539483, −18.09593883107125966084389820669, −16.63835464195437891585540718849, −14.73571689647835057769478847139, −13.8933173921017815599084530820, −12.668769108959763130106505335383, −11.345369969190425916671907977624, −9.36737334305044522412781719937, −8.40895081334319114935450569619, −6.68090083472023769975174485460, −5.45592640868744703473300792598, −2.9059847727421980431917766389, −1.49822208265687061109279131252, 1.9520464967871546869327975157, 3.92125738045100425198357315497, 5.273494634605773076838741649855, 7.198092654673264343997172931972, 8.945998351174364585446495123163, 9.98909510841826610155436269656, 11.09740926757498083893157884053, 13.21626707717390909814638059477, 14.23248852074343061922338650243, 15.2981237317562762265909484423, 16.939488056099957288882923853069, 17.65868265594300846058723622672, 19.65405032570482338954403786318, 20.59843655100138817988906801396, 21.57930938719149185461982082023, 22.65487788603621474569935750955, 24.29080798967969853806392301574, 25.576188736632627808992540384180, 26.34698299040620397252368215864, 27.55560941361184879854647722606, 28.64699601266026094374315772330, 30.09085514473229209698581535361, 31.00412380606351041606015914060, 32.64062339061599163542143235238, 33.07442184972861923944137476912

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