↑

Properties

Label 1-153-153.122-r0-0-0
Degree $1$
Conductor $153$
Sign $0.773 - 0.633i$
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  + (−0.258 − 0.965i)2-s + (−0.866 + 0.5i)4-s + (−0.991 + 0.130i)5-s + (0.991 + 0.130i)7-s + (0.707 + 0.707i)8-s + (0.382 + 0.923i)10-s + (−0.130 + 0.991i)11-s + (0.866 − 0.5i)13-s + (−0.130 − 0.991i)14-s + (0.5 − 0.866i)16-s + (0.707 − 0.707i)19-s + (0.793 − 0.608i)20-s + (0.991 − 0.130i)22-s + (0.793 + 0.608i)23-s + (0.965 − 0.258i)25-s + (−0.707 − 0.707i)26-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.5i)4-s + (−0.991 + 0.130i)5-s + (0.991 + 0.130i)7-s + (0.707 + 0.707i)8-s + (0.382 + 0.923i)10-s + (−0.130 + 0.991i)11-s + (0.866 − 0.5i)13-s + (−0.130 − 0.991i)14-s + (0.5 − 0.866i)16-s + (0.707 − 0.707i)19-s + (0.793 − 0.608i)20-s + (0.991 − 0.130i)22-s + (0.793 + 0.608i)23-s + (0.965 − 0.258i)25-s + (−0.707 − 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.773 - 0.633i$
Analytic conductor: \(0.710529\)
Root analytic conductor: \(0.710529\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (0:\ ),\ 0.773 - 0.633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8134732320 - 0.2907001989i\)
\(L(\frac12)\) \(\approx\) \(0.8134732320 - 0.2907001989i\)
\(L(1)\) \(\approx\) \(0.8138034242 - 0.2699469455i\)
\(L(1)\) \(\approx\) \(0.8138034242 - 0.2699469455i\)

Euler product

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

−27.78890368653564016881035877072, −26.96769146838586439678570472072, −26.426219238301199745948813355578, −25.00696750731155889106758978671, −24.14318203234404717532047036995, −23.56845997441586429795609873440, −22.58647359711205930205396332429, −21.24551016392770720860694352403, −20.10472495053072538276455826832, −18.773636530469164029717497844868, −18.32300717846294300471684372143, −16.79001882831435971831168941272, −16.237547209449850696913121110136, −15.09389422467373379799009947447, −14.258138004538109009691743644676, −13.1611279681775110013140520926, −11.58165943899316908876629367381, −10.7326950331603221921553093156, −9.01168698339718861571613590767, −8.207914027454654617489989251229, −7.379389504557446023608664921441, −5.97573952263019046815042652273, −4.753702455605236837568884847051, −3.65438840639338168001820460435, −1.09735086234598557062938041269, 1.27493993131866962933574189370, 2.85900330401710044636904984375, 4.14302165944065559624247496041, 5.13335603861132361838176430963, 7.35692456863901936624026624648, 8.17214065004708846384160310586, 9.305906951511597930816183118413, 10.73882359955765887546874264519, 11.41625937196838004679622184006, 12.34148048977720672372165110146, 13.48535878400494430898115968951, 14.797397192523999800247438715283, 15.75160020276369969545008456126, 17.32790284637369838994401412943, 18.09426968159336962576839874530, 19.02487646309381740723741806810, 20.18707404855358377017352833766, 20.67674607857960230340030213411, 21.88942656010695594146149206622, 23.00303697860954712236349790009, 23.61682395799043486696946815106, 25.07922570080081690498640566122, 26.31795145922360971914646093428, 27.19433130592367513476950125428, 27.95340755818288258896405586684

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