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Properties

Label 1-153-153.112-r1-0-0
Degree $1$
Conductor $153$
Sign $-0.997 - 0.0648i$
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  + (0.965 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.997 - 0.0648i$
Analytic conductor: \(16.4421\)
Root analytic conductor: \(16.4421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (1:\ ),\ -0.997 - 0.0648i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03118701896 - 0.9602166642i\)
\(L(\frac12)\) \(\approx\) \(0.03118701896 - 0.9602166642i\)
\(L(1)\) \(\approx\) \(1.077569670 - 0.4560155298i\)
\(L(1)\) \(\approx\) \(1.077569670 - 0.4560155298i\)

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 \( 1 + (0.130 - 0.991i)T \)
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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

−28.391493259384006358771143076351, −27.02473276013247964378853475480, −26.10206513048964818391262002851, −25.36135984291812662796985894110, −24.018382693409819176949757800639, −23.190771816791461559786703554724, −22.63017593732081613888115008102, −21.6559358389760629473805118473, −20.190103999840937845101554074461, −19.769872568426231838875083534526, −18.33825790719647353278953807947, −16.92721646300853088160734598577, −15.98874477935375774007972423694, −15.06564461927877767099356042951, −14.2693448119362441419859546157, −12.87440289357675152493469571406, −12.286837579396462342703101901930, −10.90030901009363447907974954121, −10.019126888025963103140002548, −7.89248633213317105117517243205, −7.2338833331924368310988026744, −6.11303378363230487008353285209, −4.594360437032709035035353282564, −3.59559313424640685562980735163, −2.43268217239424029500834792932, 0.23710784431544284574517152389, 2.30989239034095777277805555236, 3.539098799906989556434865572458, 4.73267629917863226500426897325, 5.8378454793777582888245713823, 7.09150290261224213314239713043, 8.47820957005032922467814628017, 9.83600063208837655683061652097, 11.19907372763577974926761318929, 12.15349263945176225762593334177, 12.8719350101517375994318054183, 13.967023203334862888972953390064, 15.34789383667254114650180653936, 15.89317590697255929703207470643, 16.90063544546223476256712438534, 18.87615081253769857617714608480, 19.43137398305882111710382220665, 20.42499343378725284472481037512, 21.60026881444815302002574303708, 22.23205562603910517781882826445, 23.53356260974615082442657255134, 24.03572420246799644259245151125, 25.04990139899792928384553527831, 26.19363539774605792551950738162, 27.493400624768326630244261104146

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