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Properties

Label 1-153-153.104-r1-0-0
Degree $1$
Conductor $153$
Sign $0.133 + 0.991i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.133 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.133 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.133 + 0.991i$
Analytic conductor: \(16.4421\)
Root analytic conductor: \(16.4421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (1:\ ),\ 0.133 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.212266218 + 1.934443494i\)
\(L(\frac12)\) \(\approx\) \(2.212266218 + 1.934443494i\)
\(L(1)\) \(\approx\) \(1.620936543 + 0.7207802631i\)
\(L(1)\) \(\approx\) \(1.620936543 + 0.7207802631i\)

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

−27.79879663863715783887894097643, −26.86489281787507494010424200312, −25.25509623502611436525607620561, −24.36184073459540247515914319509, −23.51728123241874873390198374828, −22.589641639745033466068804590431, −21.79993957670440535113527325331, −20.5216718201839889224307415435, −19.924187426890430932003845449680, −18.83531363406225120079900979787, −17.75842006843718802214660967330, −16.16661287333566998382996772798, −15.0631082179336011853221162088, −14.59997900311599543653835682487, −13.24965903589167821202132970506, −12.03529990268774483222276887132, −11.39607860786312450875825484509, −10.45228504515983200150616680990, −8.80542535521314255906904474936, −7.501579288627620838848828563512, −6.23892357889258505652053277778, −4.81367818506680757380820608441, −3.92150926059061505608007007310, −2.57378017092832365170150701060, −0.941544432252174846541596478642, 1.57811761314141607038018843056, 3.65665658595157164399827898339, 4.28807041628245694974749295453, 5.615073648967271889409299597578, 7.00076322547200261611861619707, 7.94220192611283077665258609079, 8.9430365901961284818897905497, 11.128433035818366691190593735741, 11.65702159103913906331897808541, 12.73024935141601208901072241864, 14.17240858968253655051935182375, 14.630588642769430325310806469635, 15.98126738402049672616496530197, 16.63284577615338912319497342969, 17.78660676670954314860419795006, 19.251680267741497819801832496006, 20.33054450243347006833477507778, 21.17112183500569401665047788891, 22.22491038270832260302040843386, 23.3617690593912335574606300448, 23.92530307798550989904602110704, 24.75221821800092080021113774250, 25.888527472903000995198820240075, 27.09405859057722759998659092174, 27.65930762894801043299514850089

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