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Properties

Label 1-555-555.353-r1-0-0
Degree $1$
Conductor $555$
Sign $0.594 - 0.804i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.594 - 0.804i$
Analytic conductor: \(59.6430\)
Root analytic conductor: \(59.6430\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (1:\ ),\ 0.594 - 0.804i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.311609972 - 2.175658539i\)
\(L(\frac12)\) \(\approx\) \(4.311609972 - 2.175658539i\)
\(L(1)\) \(\approx\) \(2.181545767 - 0.6466876654i\)
\(L(1)\) \(\approx\) \(2.181545767 - 0.6466876654i\)

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

−23.30946001840421413137398373458, −22.51189129758862097013477989361, −21.526106808547618716529686083383, −20.8497762769178397482453563318, −20.47173230588513880758623175499, −18.872523129946855837152933023841, −18.351280112326673712495168453391, −17.048812961857030747343981511524, −16.378669602834596304208120469352, −15.62511720605934893170315706320, −14.4567866016403050215125354564, −14.14235449944568765768339745132, −13.13273648394984735151891495280, −12.17570281985866691857085987143, −11.289629428417172917203838992677, −10.75381671856861432629921687258, −9.15364979251342897515399686172, −8.096387760267168572425823306159, −7.50862615348398040414235336347, −6.16942671854018904033280905625, −5.50727577230772750086376710696, −4.52988350909533526201099588877, −3.53186210084782045506950570756, −2.45379676287338286124570321151, −1.207299363219441141332734575772, 0.99625590614149689679333811000, 1.962693146266996063798945731681, 3.14669917862955152926818906907, 4.180169017760337760489867996519, 5.13574240292052444897731032198, 5.806038479122508254658964135757, 7.19129897816607058857335015941, 7.83137530239315660332171138828, 9.218055949236564249279194822292, 10.432708056579476796697413059491, 10.967628779502498089181624731648, 11.92167822750020791740413140353, 12.81453831600423686010198907632, 13.56231038506933808013679223482, 14.47674583615248995395147479550, 15.25686506756529206970247432118, 15.85735290734786935115168553949, 17.21162661008590358420187485270, 17.940627797025556014137011163598, 18.98684513481722934521747773103, 19.95611297193008325270874931960, 20.74321523460947983327167948765, 21.18815214051384157652738647831, 22.16465170534215685545124425729, 23.14863609835749599705377466653

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