Properties

Label 1-1045-1045.83-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.298 - 0.954i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (0.994 − 0.104i)2-s + (0.207 − 0.978i)3-s + (0.978 − 0.207i)4-s + (0.104 − 0.994i)6-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s i·12-s + (0.406 − 0.913i)13-s + (0.978 + 0.207i)14-s + (0.913 − 0.406i)16-s + (0.406 + 0.913i)17-s + (−0.951 − 0.309i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (−0.104 − 0.994i)24-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)2-s + (0.207 − 0.978i)3-s + (0.978 − 0.207i)4-s + (0.104 − 0.994i)6-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s i·12-s + (0.406 − 0.913i)13-s + (0.978 + 0.207i)14-s + (0.913 − 0.406i)16-s + (0.406 + 0.913i)17-s + (−0.951 − 0.309i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (−0.104 − 0.994i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.298 - 0.954i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.298 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.897627075 - 2.130597490i\)
\(L(\frac12)\) \(\approx\) \(2.897627075 - 2.130597490i\)
\(L(1)\) \(\approx\) \(2.115510677 - 0.9030734892i\)
\(L(1)\) \(\approx\) \(2.115510677 - 0.9030734892i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.994 - 0.104i)T \)
3 \( 1 + (0.207 - 0.978i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
13 \( 1 + (0.406 - 0.913i)T \)
17 \( 1 + (0.406 + 0.913i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.978 + 0.207i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.743 - 0.669i)T \)
53 \( 1 + (-0.406 + 0.913i)T \)
59 \( 1 + (-0.669 - 0.743i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (0.743 - 0.669i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.994 + 0.104i)T \)
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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

−21.51790014601477163671286699384, −20.9660219602206079435986496118, −20.63933063251044780276218338092, −19.69419241839023341820664915954, −18.74295413669796599455640647601, −17.46424285763261696797061105233, −16.742330587542044280645449894472, −16.10124113806768710883168849880, −15.27711008585846902009359113868, −14.54707191738163903313458572416, −14.0194163048737424817842258161, −13.29597250421910019936151317030, −12.048046305019331113355685037405, −11.197532426342003216503520088677, −10.95754122365896503392285812063, −9.71250124417349359627492679222, −8.85233162189810954057842284181, −7.77719744891867891318568947059, −7.0596538613644543361761433879, −5.75687770840579310340996679111, −5.112745835731640952396822188290, −4.298999518995467788386686742091, −3.64278755978612979404346221956, −2.59092633524418769442529487135, −1.53090334009249473215755144747, 1.18593520155319340999629278526, 1.92221359026734926458663406388, 2.9331054438007498633138686983, 3.78556852533371004068806033429, 5.086472331872207501512938755850, 5.69893375442793076836767951978, 6.55956964415421361733498832255, 7.58022256309537619178493735865, 8.10615772176676636754262639894, 9.1415955185429368033947518947, 10.76775126822941344526487464662, 11.03534286219493866539646467253, 12.21424487946440373142508667535, 12.65556171426920939931232790346, 13.38392554157090853801809299363, 14.37667192995779651862299395537, 14.75048382812721013065521735934, 15.57133219974416790713691131318, 16.74870424935061147409225681239, 17.52977214963780052372941300518, 18.36887932553790338912224007613, 19.15406285619186583534278750733, 19.9682361212283982426155252067, 20.69803035250166289720843559811, 21.27784840653366277905865289876

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