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Properties

Label 1-555-555.98-r1-0-0
Degree $1$
Conductor $555$
Sign $0.992 + 0.120i$
Analytic cond. $59.6430$
Root an. cond. $59.6430$
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.984 − 0.173i)7-s + (0.5 + 0.866i)8-s + (−0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.866 − 0.5i)14-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + (−0.342 − 0.939i)19-s + (−0.173 − 0.984i)22-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)26-s + (−0.642 − 0.766i)28-s + (0.866 − 0.5i)29-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.984 − 0.173i)7-s + (0.5 + 0.866i)8-s + (−0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.866 − 0.5i)14-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + (−0.342 − 0.939i)19-s + (−0.173 − 0.984i)22-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)26-s + (−0.642 − 0.766i)28-s + (0.866 − 0.5i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(59.6430\)
Root analytic conductor: \(59.6430\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (1:\ ),\ 0.992 + 0.120i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.565915226 + 0.2156951719i\)
\(L(\frac12)\) \(\approx\) \(3.565915226 + 0.2156951719i\)
\(L(1)\) \(\approx\) \(1.819969283 + 0.2617669945i\)
\(L(1)\) \(\approx\) \(1.819969283 + 0.2617669945i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (-0.984 - 0.173i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.984 - 0.173i)T \)
59 \( 1 + (0.984 - 0.173i)T \)
61 \( 1 + (-0.642 + 0.766i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.984 + 0.173i)T \)
83 \( 1 + (-0.642 - 0.766i)T \)
89 \( 1 + (-0.984 + 0.173i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−23.0291437484614830283871317880, −22.49388047251283202524882352182, −21.35140910861428753447249754302, −20.82264892652229788813475158708, −19.87289992372664208857027517478, −19.145540282703187732189846609745, −18.31974138960033102781692363074, −17.0714210762100231105669419681, −16.02952458542295028592821924760, −15.44413443111817358485135605336, −14.61250345573388117672720914732, −13.573337037621847677611236096352, −12.68784260867507229371491330919, −12.40050720045297448010890673521, −11.086277705808268845649974136689, −10.23724656671753349808365702255, −9.58105397602506465565712977006, −8.09437038143329738633612932368, −7.0607150053601390457610017687, −6.0214338685398609195503412376, −5.415701964421149860181752740750, −4.07083407669721272604353753710, −3.31842822201661048037773742370, −2.28198803859660933443157814646, −1.00090455123069411364187303128, 0.766266199533227939692266075809, 2.57178272006407473292830156611, 3.27116838841780814760827399992, 4.31841069671783967697365424501, 5.384591713910790348122528152191, 6.36611260098219305783935953895, 6.9531990524904954092316546536, 8.168965897495273701592763933018, 9.08481522137054045503845872170, 10.41398178050380206559447224834, 11.22202585054361335808810382544, 12.19324676466572446801864859666, 13.11564850787043969702579952235, 13.68425760436870435712805918717, 14.50754477612995339511202589996, 15.71312334048818899378966507131, 16.188393171801578258131106097039, 16.838814204211054486899074642975, 18.1011065842654326299520703983, 19.13476017354310319247974622599, 19.84839765459310011999190668442, 21.08982106944910711239540882657, 21.362277595161902062794813002503, 22.51441811167957975992374862696, 23.13103304288272370807649298298

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