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Properties

Label 1-2552-2552.1251-r0-0-0
Degree $1$
Conductor $2552$
Sign $-0.993 - 0.116i$
Analytic cond. $11.8514$
Root an. cond. $11.8514$
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  + (0.0448 − 0.998i)3-s + (−0.134 + 0.990i)5-s + (−0.963 + 0.266i)7-s + (−0.995 − 0.0896i)9-s + (0.858 + 0.512i)13-s + (0.983 + 0.178i)15-s + (0.309 + 0.951i)17-s + (−0.963 − 0.266i)19-s + (0.222 + 0.974i)21-s + (0.900 − 0.433i)23-s + (−0.963 − 0.266i)25-s + (−0.134 + 0.990i)27-s + (−0.691 + 0.722i)31-s + (−0.134 − 0.990i)35-s + (−0.393 + 0.919i)37-s + ⋯
L(s)  = 1  + (0.0448 − 0.998i)3-s + (−0.134 + 0.990i)5-s + (−0.963 + 0.266i)7-s + (−0.995 − 0.0896i)9-s + (0.858 + 0.512i)13-s + (0.983 + 0.178i)15-s + (0.309 + 0.951i)17-s + (−0.963 − 0.266i)19-s + (0.222 + 0.974i)21-s + (0.900 − 0.433i)23-s + (−0.963 − 0.266i)25-s + (−0.134 + 0.990i)27-s + (−0.691 + 0.722i)31-s + (−0.134 − 0.990i)35-s + (−0.393 + 0.919i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2552\)    =    \(2^{3} \cdot 11 \cdot 29\)
Sign: $-0.993 - 0.116i$
Analytic conductor: \(11.8514\)
Root analytic conductor: \(11.8514\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2552} (1251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2552,\ (0:\ ),\ -0.993 - 0.116i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001342961543 + 0.02306497787i\)
\(L(\frac12)\) \(\approx\) \(0.001342961543 + 0.02306497787i\)
\(L(1)\) \(\approx\) \(0.7622187369 - 0.03800023144i\)
\(L(1)\) \(\approx\) \(0.7622187369 - 0.03800023144i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.0448 - 0.998i)T \)
5 \( 1 + (-0.134 + 0.990i)T \)
7 \( 1 + (-0.963 + 0.266i)T \)
13 \( 1 + (0.858 + 0.512i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.963 - 0.266i)T \)
23 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (-0.691 + 0.722i)T \)
37 \( 1 + (-0.393 + 0.919i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.393 - 0.919i)T \)
53 \( 1 + (0.691 - 0.722i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.936 - 0.351i)T \)
67 \( 1 + (-0.222 - 0.974i)T \)
71 \( 1 + (0.995 - 0.0896i)T \)
73 \( 1 + (0.983 + 0.178i)T \)
79 \( 1 + (0.995 + 0.0896i)T \)
83 \( 1 + (0.550 - 0.834i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + (-0.936 + 0.351i)T \)
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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

−19.9029076966410057099587805238, −19.25315138231950859465494517519, −18.36629858222713932769013242651, −17.30428098935140865424483971040, −16.70514623883347446716988866299, −16.29671578211756247018498767582, −15.53210232697011189743127467489, −15.09643840519739717335046791801, −13.90954436379947856534348918007, −13.3607045474404016095234769264, −12.619831562615142831655194338070, −11.85192319050476344238267385579, −10.92345226197641258222362551245, −10.35700747104107290748584948739, −9.38910257928874630265753913633, −9.09234827528028967806031856240, −8.26933112896112019619702013791, −7.37639063691762991527783114937, −6.26247239466672673508988058023, −5.53458946977593223070600995845, −4.87821138925772499107736214157, −3.88247376602505522821206260352, −3.49217951919843157959800474245, −2.44287797149456612542589297254, −1.06308274321479748038784100398, 0.00792957008434058799549796205, 1.48599211251485996873517383286, 2.25332978879967553660962283590, 3.2555523488004143429120819556, 3.60828490882536017842330509865, 5.04950796016041280459909014584, 6.21285661741856757941738184224, 6.5052403671535445704507497627, 7.01336215833157701940095295392, 8.143929821151269758763120589229, 8.647968471921534122173484299625, 9.59573574911403640304955422073, 10.59687907326055304400085748054, 11.079193388845491659509840757154, 11.99786415833944665722879563300, 12.64149005423295425409656665850, 13.37064900483779448102804229029, 13.88073682986044801470801247646, 15.02083228986972814522690381149, 15.1112738799384506429382902291, 16.43483546464879895263646274952, 16.90845686958700079996731911101, 17.89056785931953307393689340693, 18.56974294503631125486781950332, 18.94550267226084120524398157065

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