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Properties

Label 1-153-153.139-r1-0-0
Degree $1$
Conductor $153$
Sign $0.633 + 0.773i$
Analytic cond. $16.4421$
Root an. cond. $16.4421$
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  + (0.258 + 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

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

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.633 + 0.773i$
Analytic conductor: \(16.4421\)
Root analytic conductor: \(16.4421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (1:\ ),\ 0.633 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.337123545 + 1.106515634i\)
\(L(\frac12)\) \(\approx\) \(2.337123545 + 1.106515634i\)
\(L(1)\) \(\approx\) \(1.425680983 + 0.5899998650i\)
\(L(1)\) \(\approx\) \(1.425680983 + 0.5899998650i\)

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 \( 1 + (0.608 + 0.793i)T \)
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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

−27.949356077329342546460684678565, −26.83229414508531196409961517598, −25.761799901270109096063360618537, −24.603182779894214352798414131663, −23.49899388424362282886157745436, −22.553414585282044155912863222498, −21.48207990062052046934277107103, −20.830513599699672479056996694974, −20.042686934757308141861168370302, −18.506500430834160941554276242892, −17.968683617349710263539927186219, −17.00290096340449023951073613402, −15.189225624035096995490982575855, −14.10954507684807407963467977976, −13.54461956998422835300316337950, −12.198294617118161453172053909, −11.258636145522736041100140667905, −10.13765277178840151943525019230, −9.337818758840602155770095320087, −7.967412130880126974144867221753, −6.18915761398630559215952169889, −5.05383842070229744195350749801, −3.90163914030916220365345179380, −2.19244414682983401533965392479, −1.374301285653665590436540452340, 1.128226729400784246419378162706, 3.10508157696752057835710019506, 4.76649226428764850716745695379, 5.665752332454902369468525889846, 6.64195321854724012106176094324, 8.20663276834173942650017575794, 8.82638578780497403203943518329, 10.25230770052483187988518852461, 11.6270135515533292949813051346, 13.102044097488498496578505932162, 13.86859072853949282716158013912, 14.65368651394630270348234416394, 15.94057794852803394722907267173, 16.815523936428416131724812067504, 18.02514826657088295134402538199, 18.26598764294831787936470469085, 20.19748529283976416430376428632, 21.462000626610840483800713688784, 21.84538815379079493008948579807, 23.20436948017886975293230669503, 24.26358817587221657733042701397, 24.81678305667540918851945790326, 25.795092109649113371990596056061, 26.74571871749782989658918438430, 27.69757354511415494074839226902

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