Properties

Label 1-33e2-1089.250-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.00576 + 0.999i$
Analytic cond. $117.029$
Root an. cond. $117.029$
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  + (−0.345 − 0.938i)2-s + (−0.761 + 0.647i)4-s + (−0.595 + 0.803i)5-s + (0.625 − 0.780i)7-s + (0.870 + 0.491i)8-s + (0.959 + 0.281i)10-s + (0.851 + 0.524i)13-s + (−0.948 − 0.318i)14-s + (0.161 − 0.986i)16-s + (−0.516 + 0.856i)17-s + (−0.0855 + 0.996i)19-s + (−0.0665 − 0.997i)20-s + (0.0475 + 0.998i)23-s + (−0.290 − 0.956i)25-s + (0.198 − 0.980i)26-s + ⋯
L(s)  = 1  + (−0.345 − 0.938i)2-s + (−0.761 + 0.647i)4-s + (−0.595 + 0.803i)5-s + (0.625 − 0.780i)7-s + (0.870 + 0.491i)8-s + (0.959 + 0.281i)10-s + (0.851 + 0.524i)13-s + (−0.948 − 0.318i)14-s + (0.161 − 0.986i)16-s + (−0.516 + 0.856i)17-s + (−0.0855 + 0.996i)19-s + (−0.0665 − 0.997i)20-s + (0.0475 + 0.998i)23-s + (−0.290 − 0.956i)25-s + (0.198 − 0.980i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00576 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00576 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.00576 + 0.999i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (250, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.00576 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6941580229 + 0.6901644605i\)
\(L(\frac12)\) \(\approx\) \(0.6941580229 + 0.6901644605i\)
\(L(1)\) \(\approx\) \(0.7890497528 - 0.09392962083i\)
\(L(1)\) \(\approx\) \(0.7890497528 - 0.09392962083i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.345 - 0.938i)T \)
5 \( 1 + (-0.595 + 0.803i)T \)
7 \( 1 + (0.625 - 0.780i)T \)
13 \( 1 + (0.851 + 0.524i)T \)
17 \( 1 + (-0.516 + 0.856i)T \)
19 \( 1 + (-0.0855 + 0.996i)T \)
23 \( 1 + (0.0475 + 0.998i)T \)
29 \( 1 + (0.683 + 0.730i)T \)
31 \( 1 + (-0.432 - 0.901i)T \)
37 \( 1 + (0.610 + 0.791i)T \)
41 \( 1 + (0.969 - 0.244i)T \)
43 \( 1 + (-0.580 + 0.814i)T \)
47 \( 1 + (-0.532 + 0.846i)T \)
53 \( 1 + (0.774 - 0.633i)T \)
59 \( 1 + (0.272 + 0.962i)T \)
61 \( 1 + (-0.640 - 0.768i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (-0.921 - 0.389i)T \)
73 \( 1 + (0.998 + 0.0570i)T \)
79 \( 1 + (0.217 - 0.976i)T \)
83 \( 1 + (-0.964 + 0.263i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.595 - 0.803i)T \)
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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

−21.027021175021003484921502527819, −20.10742165016531308944836016864, −19.47889066282141959126004576105, −18.3627877486645732189761238159, −18.02929627376914832465009449494, −17.11522925054431516489553196720, −16.201689746030579051437884930963, −15.66057098170010543110731082837, −15.10821051223190278612287489156, −14.12300558162146682913894428193, −13.26518831620083543401328707775, −12.492851822189045769584287331418, −11.468558249717574118943973590530, −10.72484540401287141422953742482, −9.45517075446977521469155077399, −8.70871877719851590300836323807, −8.34447564702936177533824468167, −7.39420918012413741194797365610, −6.44436296083118590710091278058, −5.414312046881852729414610291633, −4.84138099292175737657139871169, −3.99175845146713738602234022485, −2.48895358939834876201182705484, −1.11306618917462727790764453237, −0.27913798850037533706707478712, 1.1296738806638673252575071181, 1.927212933304417276778115950356, 3.20200241721967283709947390195, 3.923437001901534759902827476227, 4.52267592157055336638139508857, 5.99712000786968095675477785830, 7.09009182596055365500366111706, 7.93745145254241253674635051050, 8.48643997655503037589496807874, 9.6883245901572653949567213891, 10.483553469107327544657048459063, 11.17882413932573197354390693868, 11.55451905477888936348128357733, 12.66253929446807636642000018180, 13.53520688361571387801580573447, 14.2524720707067050413229448337, 14.999833928261522675843311202185, 16.17109643948507937946434485160, 16.91943264809961077341853083378, 17.877721904476597659572648846141, 18.33694838095035046394419061650, 19.26333212141090475362702077696, 19.770321448372423395908987343993, 20.62532487666345046170491980986, 21.32944459565343562065213481548

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