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Properties

Label 1-555-555.389-r0-0-0
Degree $1$
Conductor $555$
Sign $0.328 - 0.944i$
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  + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.766 − 0.642i)7-s + (0.866 − 0.5i)8-s + (−0.5 − 0.866i)11-s + (0.342 + 0.939i)13-s + (−0.866 − 0.5i)14-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + (0.984 + 0.173i)19-s + (−0.642 − 0.766i)22-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)26-s + (−0.939 − 0.342i)28-s + (0.866 − 0.5i)29-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.766 − 0.642i)7-s + (0.866 − 0.5i)8-s + (−0.5 − 0.866i)11-s + (0.342 + 0.939i)13-s + (−0.866 − 0.5i)14-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + (0.984 + 0.173i)19-s + (−0.642 − 0.766i)22-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)26-s + (−0.939 − 0.342i)28-s + (0.866 − 0.5i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.328 - 0.944i$
Analytic conductor: \(2.57740\)
Root analytic conductor: \(2.57740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (0:\ ),\ 0.328 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.933677573 - 1.374820903i\)
\(L(\frac12)\) \(\approx\) \(1.933677573 - 1.374820903i\)
\(L(1)\) \(\approx\) \(1.701030031 - 0.5559063046i\)
\(L(1)\) \(\approx\) \(1.701030031 - 0.5559063046i\)

Euler product

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

−23.25865115056707161405294551304, −22.80296031007150432303568425676, −21.86155282363215425452709465279, −21.29539383955107582260912702450, −20.079161225304781587845599302586, −19.77434384541129686909001296364, −18.38831957769005236474178393435, −17.61152031699977010626579688989, −16.434704614308505068015447044983, −15.625251679197790627356978465681, −15.21407506296405274837242032204, −14.109033798546183110301042761590, −13.17462112227136742916171198219, −12.48672719853049086069831235944, −11.881877930904584881953903023958, −10.5648918396074170509812364913, −9.89276299697290974923822647091, −8.47109374655951715899663389479, −7.56996416947712770976529331397, −6.5511130710072800542166609924, −5.6688109127828512452594660544, −4.93196834076130542689853024796, −3.580170581443515283304174854029, −2.90255091080740407051991258091, −1.68142183204617627198513826375, 0.91922435453819423757942641762, 2.44386842039298628865771081064, 3.38890900240129479146746887397, 4.21206786873948260653836450893, 5.35648000391579358906592760253, 6.28074866466528159343012007927, 7.07164219727603931606757284005, 8.112047504527131514094461674262, 9.58421133947775166662307941403, 10.298023931682044399608654681723, 11.37877917223720955192384898255, 11.992178426626485815292360031199, 13.1592087679315142288786286842, 13.78595827123521422242748543505, 14.30147936812840670989586731713, 15.77048281923190713422340020848, 16.14227896112126218846354657222, 16.90680618131197691920620168269, 18.46119260342110682198648407354, 19.079334034750623458323955424852, 20.0974519129743392587668416092, 20.71136880352690053633670769484, 21.58948683753761099173067948935, 22.4180540718451245153568131737, 23.09016231380002060676681878177

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