↑

Properties

Label 1-569-569.175-r0-0-0
Degree $1$
Conductor $569$
Sign $0.563 + 0.826i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.730 − 0.683i)2-s + (−0.912 + 0.408i)3-s + (0.0663 − 0.997i)4-s + (0.325 + 0.945i)5-s + (−0.387 + 0.921i)6-s + (0.598 + 0.801i)7-s + (−0.633 − 0.773i)8-s + (0.666 − 0.745i)9-s + (0.883 + 0.467i)10-s + (0.219 + 0.975i)11-s + (0.346 + 0.937i)12-s + (−0.964 + 0.262i)13-s + (0.984 + 0.176i)14-s + (−0.683 − 0.730i)15-s + (−0.991 − 0.132i)16-s + (0.999 − 0.0442i)17-s + ⋯
L(s)  = 1  + (0.730 − 0.683i)2-s + (−0.912 + 0.408i)3-s + (0.0663 − 0.997i)4-s + (0.325 + 0.945i)5-s + (−0.387 + 0.921i)6-s + (0.598 + 0.801i)7-s + (−0.633 − 0.773i)8-s + (0.666 − 0.745i)9-s + (0.883 + 0.467i)10-s + (0.219 + 0.975i)11-s + (0.346 + 0.937i)12-s + (−0.964 + 0.262i)13-s + (0.984 + 0.176i)14-s + (−0.683 − 0.730i)15-s + (−0.991 − 0.132i)16-s + (0.999 − 0.0442i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.563 + 0.826i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.563 + 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.271272722 + 0.6715808346i\)
\(L(\frac12)\) \(\approx\) \(1.271272722 + 0.6715808346i\)
\(L(1)\) \(\approx\) \(1.205272865 + 0.08065277743i\)
\(L(1)\) \(\approx\) \(1.205272865 + 0.08065277743i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.730 - 0.683i)T \)
3 \( 1 + (-0.912 + 0.408i)T \)
5 \( 1 + (0.325 + 0.945i)T \)
7 \( 1 + (0.598 + 0.801i)T \)
11 \( 1 + (0.219 + 0.975i)T \)
13 \( 1 + (-0.964 + 0.262i)T \)
17 \( 1 + (0.999 - 0.0442i)T \)
19 \( 1 + (-0.997 + 0.0663i)T \)
23 \( 1 + (0.873 - 0.487i)T \)
29 \( 1 + (-0.0442 + 0.999i)T \)
31 \( 1 + (-0.980 + 0.197i)T \)
37 \( 1 + (-0.801 + 0.598i)T \)
41 \( 1 + (0.487 + 0.873i)T \)
43 \( 1 + (-0.730 - 0.683i)T \)
47 \( 1 + (0.999 + 0.0221i)T \)
53 \( 1 + (0.387 - 0.921i)T \)
59 \( 1 + (-0.999 - 0.0221i)T \)
61 \( 1 + (0.839 + 0.544i)T \)
67 \( 1 + (0.999 + 0.0442i)T \)
71 \( 1 + (-0.633 + 0.773i)T \)
73 \( 1 + (0.997 - 0.0663i)T \)
79 \( 1 + (-0.699 + 0.714i)T \)
83 \( 1 + (0.958 + 0.283i)T \)
89 \( 1 + (0.980 + 0.197i)T \)
97 \( 1 + (0.616 + 0.787i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.47871370087031594318335449180, −22.467269906414162987350630821363, −21.42385391830502938605646192046, −21.14856296382841486717734956670, −19.91239088721596175022661495868, −18.851464465549809780384945323286, −17.55108026366577730015335919819, −16.99734807971629372969108955574, −16.76153068654806408785830419302, −15.70117860787400633543008453211, −14.51884540395865183206258201997, −13.69975290935947327938906292745, −12.95880041108943388020835002983, −12.244921966035187871760054041358, −11.3944872817177644587697601534, −10.43357844727690805789699656642, −9.048255284360571367975956005620, −7.92403700093706642235402622997, −7.34040723138461213988402119077, −6.155353754238617496727051353865, −5.400586346259703119549695389348, −4.73311818574483718645230381096, −3.7379876784681407392490812578, −2.03108149204309170434024780718, −0.657832107813152773164319640546, 1.5880423173272533763096466049, 2.50331627663622882631126393590, 3.702537403199838726545414953656, 4.86996969469422840037215059179, 5.3810809091990873595599424675, 6.4899505211908488449379261233, 7.19685878571550961457324282478, 9.08697191942390634000834817966, 9.97684934961762361556007977505, 10.60761962908520242299366549852, 11.46115870900016993849583553382, 12.22073052840514903708927862684, 12.80232776969406804465625693025, 14.41646635794261040178030114556, 14.77177741827907798432315005690, 15.401307007452879244614583081982, 16.77005238245584445529855715735, 17.667704430377984259801808330683, 18.49138861916481042729568184453, 19.06586580906793394852992012347, 20.35251586486722045429961459604, 21.29263991104662684453418396660, 21.741682054360225152157488357399, 22.41898778859489147319264063306, 23.14049845990568952929900715793

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