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Properties

Label 1-1045-1045.103-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.577 + 0.816i$
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  + (0.743 + 0.669i)2-s + (0.994 + 0.104i)3-s + (0.104 + 0.994i)4-s + (0.669 + 0.743i)6-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (0.978 + 0.207i)9-s + i·12-s + (−0.207 + 0.978i)13-s + (−0.104 + 0.994i)14-s + (−0.978 + 0.207i)16-s + (0.207 + 0.978i)17-s + (0.587 + 0.809i)18-s + (0.5 + 0.866i)21-s + (−0.866 − 0.5i)23-s + (−0.669 + 0.743i)24-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.994 + 0.104i)3-s + (0.104 + 0.994i)4-s + (0.669 + 0.743i)6-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (0.978 + 0.207i)9-s + i·12-s + (−0.207 + 0.978i)13-s + (−0.104 + 0.994i)14-s + (−0.978 + 0.207i)16-s + (0.207 + 0.978i)17-s + (0.587 + 0.809i)18-s + (0.5 + 0.866i)21-s + (−0.866 − 0.5i)23-s + (−0.669 + 0.743i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.549009423 + 2.991759155i\)
\(L(\frac12)\) \(\approx\) \(1.549009423 + 2.991759155i\)
\(L(1)\) \(\approx\) \(1.722919946 + 1.363111690i\)
\(L(1)\) \(\approx\) \(1.722919946 + 1.363111690i\)

Euler product

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

−21.00293525677813734551194126250, −20.53607002836138370296332152477, −19.90191038854995717315801102666, −19.34737602465041502256333665851, −18.22292932656546682580312968158, −17.76169011071771744764884262112, −16.24680890909933199625171327814, −15.57624545892646272969787791578, −14.48795346925401621742057933336, −14.29219918502898810957874469001, −13.37377185489746966385807721388, −12.765395750246215819693438002915, −11.83232354952891233995162155656, −10.87605644866995768756330167410, −10.10750107589696953022557460343, −9.44808266869213743612931450611, −8.28737247698707045927514353221, −7.47813906423022604117903713043, −6.63864851261955349920076088209, −5.280431076168753229303205152685, −4.60194987933916191239253303270, −3.55387589262663413671741873067, −2.97972766296919797762629877097, −1.84995045524783738084360051048, −0.981080647316015020349926944031, 1.986725742854691271841565985696, 2.41257841772765238092121527305, 3.81537603305572635625358664716, 4.24699762521226820986769916758, 5.38159728247242642786122732351, 6.23304330769033357544490620959, 7.2812688395544932354029254987, 8.04127706369911970693527402656, 8.713243448959226365989732980263, 9.43295844668135183947286835837, 10.68146965186772672650794931793, 11.85014018890068295830355118463, 12.40387144629352988233880285578, 13.37189147583824697111159851196, 14.12958769837344734787828951618, 14.709728353737544760392068839962, 15.32236081413886862708882143733, 16.06554893827353041838358860970, 16.94437701164067749924899175842, 17.87052968785215280530946246285, 18.76717989756043776938282664441, 19.466762053166847922088550092257, 20.64290375182513820519357233908, 21.074887487506595927130495581604, 21.80508384595061220693079115529

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