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Properties

Label 1-555-555.248-r0-0-0
Degree $1$
Conductor $555$
Sign $0.940 + 0.339i$
Analytic cond. $2.57740$
Root an. cond. $2.57740$
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 + (−0.866 + 0.5i)7-s − i·8-s − 11-s + (0.866 − 0.5i)13-s + 14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.866 + 0.5i)22-s − i·23-s − 26-s + (−0.866 − 0.5i)28-s + 29-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)7-s − i·8-s − 11-s + (0.866 − 0.5i)13-s + 14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.866 + 0.5i)22-s − i·23-s − 26-s + (−0.866 − 0.5i)28-s + 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.940 + 0.339i$
Analytic conductor: \(2.57740\)
Root analytic conductor: \(2.57740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (0:\ ),\ 0.940 + 0.339i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6810541359 + 0.1191949025i\)
\(L(\frac12)\) \(\approx\) \(0.6810541359 + 0.1191949025i\)
\(L(1)\) \(\approx\) \(0.6460747740 - 0.04380886095i\)
\(L(1)\) \(\approx\) \(0.6460747740 - 0.04380886095i\)

Euler product

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

−23.54528702246736604685909290721, −22.72136960041311322184205768642, −21.5056069873483261751777898476, −20.58282206069775550880978947370, −19.68273089187546323783335167538, −19.14552016444827885902466371753, −18.11790079425346333807595077327, −17.503379033932799780689847385089, −16.49586628067650296026446373566, −15.721781656568243639257518967863, −15.33086767270872575609124832156, −13.719577549352352438363963861961, −13.44332614606796148638450795818, −11.97158287919235494076421836377, −10.87875234589873903359826258515, −10.297828705925125392189314242610, −9.291354265769523027169195324590, −8.53396486055709436381128990958, −7.459287035296553976317342414582, −6.684929036622915753629438709619, −5.85123419966246677528341187919, −4.66863293842934470186195060304, −3.27551449716060310347317358591, −2.06140177978882615371284242468, −0.61537170743746852862941730615, 0.95978776831742052480854789608, 2.53948981060773271658957378768, 3.060117742562540454649544737408, 4.40662428114931665212813920314, 5.89663005837865019754788928663, 6.71050341565040826752996208628, 7.9351545780995466174847937661, 8.58294668645459961244076405686, 9.60809823587656070287319151578, 10.33966690006987146070140223310, 11.16284670360997500702695174290, 12.24094372650145251096955569181, 12.89825118012252861844210477173, 13.752248310850051864011988784132, 15.35353452127723877750119594998, 15.94547451242228723509736675323, 16.55752058935905577013696484223, 17.922043664071443190301577745157, 18.281520473807599659751060769742, 19.11702383622369923919874783164, 20.00790627717068372682557158798, 20.76159258856341491770848151689, 21.47589154601973302678776932571, 22.54255911988406333836817685765, 23.1069572410840581745600745382

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