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Properties

Label 1-153-153.32-r1-0-0
Degree $1$
Conductor $153$
Sign $-0.582 + 0.812i$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.965 + 0.258i)5-s + (−0.965 + 0.258i)7-s + i·8-s + (0.707 + 0.707i)10-s + (−0.965 + 0.258i)11-s + (0.5 + 0.866i)13-s + (−0.965 − 0.258i)14-s + (−0.5 + 0.866i)16-s + i·19-s + (0.258 + 0.965i)20-s + (−0.965 − 0.258i)22-s + (0.258 − 0.965i)23-s + (0.866 + 0.5i)25-s + i·26-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.965 + 0.258i)5-s + (−0.965 + 0.258i)7-s + i·8-s + (0.707 + 0.707i)10-s + (−0.965 + 0.258i)11-s + (0.5 + 0.866i)13-s + (−0.965 − 0.258i)14-s + (−0.5 + 0.866i)16-s + i·19-s + (0.258 + 0.965i)20-s + (−0.965 − 0.258i)22-s + (0.258 − 0.965i)23-s + (0.866 + 0.5i)25-s + i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.582 + 0.812i$
Analytic conductor: \(16.4421\)
Root analytic conductor: \(16.4421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (1:\ ),\ -0.582 + 0.812i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.299418904 + 2.530995725i\)
\(L(\frac12)\) \(\approx\) \(1.299418904 + 2.530995725i\)
\(L(1)\) \(\approx\) \(1.470535724 + 0.9922117462i\)
\(L(1)\) \(\approx\) \(1.470535724 + 0.9922117462i\)

Euler product

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

−27.83450106482466366744314892702, −26.134207081867612634155616106832, −25.43632792148482948675315209599, −24.34040156073007722730925866446, −23.34597302966179096410233522533, −22.43250410315879748362498848622, −21.53083718038706342185494397025, −20.680950733712668186083671715397, −19.76836444179893833327834265810, −18.65224810793448798387289367649, −17.48803156250753405609593141888, −16.09229619832326167114312993266, −15.342037948110037433600903462154, −13.7653448626549660823674794333, −13.28573313702419928155845984619, −12.46937722593932479625818223307, −10.858267266476888638885748707750, −10.15281173670462722451541873774, −9.02369006217330594674056431920, −7.11483261830555446832125841474, −5.90085806080358521586221411599, −5.1349023184569807445745312764, −3.46764567168427901861653071011, −2.45086964915936914734938698012, −0.77650500981693978486878221637, 2.15344724865663405312563206059, 3.24755610749192288661437068383, 4.773581951423422741824226477875, 6.05399587781757063582072418559, 6.62906852283956032026717797571, 8.12621727299931723627549660519, 9.5228126358665989832689263379, 10.66354502971722979169716381744, 12.14450148661830819436043005062, 13.113096576828574118482995208625, 13.832170975470903506977978917504, 14.93939393277973853718186189402, 16.045218865822629916618310546977, 16.81399363958803572566009155708, 18.06305332490208812453785803727, 19.077625509895605322617740985794, 20.79416537673767940278427932477, 21.20379462558745181885389422465, 22.45955481406329834877288576011, 22.98060557334078867058258126290, 24.23612267201775141818646077067, 25.213158260848536467617061100966, 25.9551468763955339204524427576, 26.57591190069243370100781089992, 28.611291738319106490437148710185

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