↑

Properties

Label 1-153-153.56-r0-0-0
Degree $1$
Conductor $153$
Sign $-0.310 + 0.950i$
Analytic cond. $0.710529$
Root an. cond. $0.710529$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (−0.793 + 0.608i)5-s + (0.793 + 0.608i)7-s + (−0.707 − 0.707i)8-s + (0.923 − 0.382i)10-s + (−0.608 + 0.793i)11-s + (−0.866 − 0.5i)13-s + (−0.608 − 0.793i)14-s + (0.5 + 0.866i)16-s + (−0.707 + 0.707i)19-s + (−0.991 + 0.130i)20-s + (0.793 − 0.608i)22-s + (−0.991 − 0.130i)23-s + (0.258 − 0.965i)25-s + (0.707 + 0.707i)26-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (−0.793 + 0.608i)5-s + (0.793 + 0.608i)7-s + (−0.707 − 0.707i)8-s + (0.923 − 0.382i)10-s + (−0.608 + 0.793i)11-s + (−0.866 − 0.5i)13-s + (−0.608 − 0.793i)14-s + (0.5 + 0.866i)16-s + (−0.707 + 0.707i)19-s + (−0.991 + 0.130i)20-s + (0.793 − 0.608i)22-s + (−0.991 − 0.130i)23-s + (0.258 − 0.965i)25-s + (0.707 + 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.310 + 0.950i$
Analytic conductor: \(0.710529\)
Root analytic conductor: \(0.710529\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (0:\ ),\ -0.310 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2621388728 + 0.3612847327i\)
\(L(\frac12)\) \(\approx\) \(0.2621388728 + 0.3612847327i\)
\(L(1)\) \(\approx\) \(0.5355083425 + 0.1494253790i\)
\(L(1)\) \(\approx\) \(0.5355083425 + 0.1494253790i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (-0.793 + 0.608i)T \)
7 \( 1 + (0.793 + 0.608i)T \)
11 \( 1 + (-0.608 + 0.793i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.991 - 0.130i)T \)
29 \( 1 + (0.130 + 0.991i)T \)
31 \( 1 + (0.608 + 0.793i)T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (-0.130 + 0.991i)T \)
43 \( 1 + (-0.258 + 0.965i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (0.965 - 0.258i)T \)
61 \( 1 + (-0.793 - 0.608i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.382 - 0.923i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (0.608 - 0.793i)T \)
83 \( 1 + (0.965 + 0.258i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.130 - 0.991i)T \)
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

−27.614607653477301526021785944172, −26.764358069621551000737526097054, −26.14162224565797903796421141336, −24.50039933732220408333143053788, −24.15173792869199781063028933698, −23.28785311265713569553893505616, −21.467018522945800817534880198759, −20.5750507734545585937573209727, −19.62641828137012729457753776397, −18.88984981279954660001566410321, −17.5630283322050393349747354892, −16.82522205574677007827126314213, −15.868741328506984884559616220893, −14.91581379343919920779409962122, −13.64610151696414877489036206808, −12.02099084675325578206031430721, −11.22314518155346784224646627465, −10.1510935084684619548052662214, −8.75951893484178054998927864867, −7.98410345879398647358683532384, −7.08755805437929675739363287773, −5.46683544286590297785257294387, −4.16623661973628478569522038350, −2.215724562375827269899063204584, −0.498150743193475621234058690821, 1.93660390286094739436625894172, 3.09295050644652226064231286040, 4.77854837871689770921185050502, 6.53918904872511499109705078217, 7.77604131421782882226620322568, 8.30304391050368036508150626883, 9.901293865704906650072501681773, 10.72539024281267428194867494767, 11.876914189883273604098911774859, 12.52561489988497400672291955013, 14.645031119617573733724139352861, 15.263972384661615403548563579657, 16.342368646387978593957013374988, 17.742138827345846344438751739955, 18.24882008829831878093642005081, 19.32802951031475086512047051783, 20.18720982498809356179410205231, 21.22539878165638056522496176323, 22.26444867855812194819803007425, 23.5340670196376712935479723736, 24.59185608643578371148354370191, 25.580608500697618395536789473879, 26.52543810275556298754892595750, 27.49231638239691287927710997118, 27.93409504590375726132248766101

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