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Properties

Label 1-1045-1045.388-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.537 + 0.843i$
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  + (−0.207 − 0.978i)2-s + (0.406 − 0.913i)3-s + (−0.913 + 0.406i)4-s + (−0.978 − 0.207i)6-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + i·12-s + (0.743 − 0.669i)13-s + (0.913 + 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.743 − 0.669i)17-s + (−0.587 + 0.809i)18-s + (0.5 + 0.866i)21-s + (−0.866 − 0.5i)23-s + (0.978 − 0.207i)24-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (0.406 − 0.913i)3-s + (−0.913 + 0.406i)4-s + (−0.978 − 0.207i)6-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + i·12-s + (0.743 − 0.669i)13-s + (0.913 + 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.743 − 0.669i)17-s + (−0.587 + 0.809i)18-s + (0.5 + 0.866i)21-s + (−0.866 − 0.5i)23-s + (0.978 − 0.207i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.537 + 0.843i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.537 + 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2620367032 - 0.4779185570i\)
\(L(\frac12)\) \(\approx\) \(-0.2620367032 - 0.4779185570i\)
\(L(1)\) \(\approx\) \(0.5292640215 - 0.5594446015i\)
\(L(1)\) \(\approx\) \(0.5292640215 - 0.5594446015i\)

Euler product

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

−22.06976170764610501290012561464, −21.50804070348278330956770757391, −20.34853259276560807712123102963, −19.72865080337364497409866321817, −18.98961840049346750198943966043, −17.96810664596756749407219012787, −17.081187403810402431702950498875, −16.44729912185767973108798394777, −15.88051316718193003429766864816, −15.15858859867564070835245386942, −14.29530483253118815988661197604, −13.609090002085594683181630652826, −13.04925217140111504556921582610, −11.4970994695009592610068312707, −10.52468931596416538430745924219, −9.91980575618480433831532098235, −9.122973154988013898275531178849, −8.38948310767101629062232438813, −7.55593128909045837275696780577, −6.48722244138857478514689098378, −5.8671048687788761846418330280, −4.52133888480704207016567699940, −4.12991991336111103237479258951, −3.15419059623843721575261654072, −1.5271432979385501952847869355, 0.23599237083766466386472012410, 1.489804907174227117864078672708, 2.50650238116396044413446888865, 3.04369353329089854099568192561, 4.09835365110235946395329728150, 5.42616750782543793043010359960, 6.28841096341314255852783826794, 7.34529121661961895269379029219, 8.45103707509079429720225570067, 8.78409396487679854362273006424, 9.74122221180939435115002231931, 10.69240286159558397733668206625, 11.70280590293182084246964773605, 12.26929027040967267861864241484, 13.00929912679882581250111367714, 13.61783785198854772988241726115, 14.42739301688641003519100440606, 15.514807108031126293153349156543, 16.394996812087347079975434379969, 17.7313100742300899862791237461, 18.058539472903690774517720026289, 18.72117423849834613321178864081, 19.57614871328059287974094271269, 20.03077240537905054095198607622, 20.85588916196066320012481073704

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