Properties

Label 1-1045-1045.619-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.653 - 0.756i$
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  + (0.978 − 0.207i)2-s + (−0.913 − 0.406i)3-s + (0.913 − 0.406i)4-s + (−0.978 − 0.207i)6-s + (0.809 + 0.587i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s − 12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.669 + 0.743i)17-s + (0.809 + 0.587i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)24-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)2-s + (−0.913 − 0.406i)3-s + (0.913 − 0.406i)4-s + (−0.978 − 0.207i)6-s + (0.809 + 0.587i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s − 12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.669 + 0.743i)17-s + (0.809 + 0.587i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.653 - 0.756i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.653 - 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.165900442 - 0.9908620611i\)
\(L(\frac12)\) \(\approx\) \(2.165900442 - 0.9908620611i\)
\(L(1)\) \(\approx\) \(1.578209529 - 0.4249998912i\)
\(L(1)\) \(\approx\) \(1.578209529 - 0.4249998912i\)

Euler product

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

−21.57159627567499585384388196518, −21.30123885033740934657985748198, −20.32976943049754525728961173656, −19.608090451587636186568494002748, −18.30013094610894314758814860723, −17.35953587544852470054549026315, −17.01351787124237282404414094786, −15.987202521273974005194211644526, −15.51492409206130268272610607269, −14.41521636643466453572129489241, −13.96229433559571408061756246087, −12.878535345591345312905346987102, −12.05932942614542191091895391686, −11.36313635747862353097237151619, −10.84649655717124604384681910283, −9.86795389650773540234026391967, −8.71792447130352125112110613982, −7.304275695332220673684323378588, −7.02158882521171102709493994630, −5.90770390254476514447349452786, −4.94935967903385293527575074124, −4.564100106346334042153711968301, −3.65265369103314321247056629488, −2.35545993588463815431362367505, −1.15605775473305443141488732183, 0.9630183330027035329290229101, 2.07643208622400189833249623653, 2.834855079034095031810247813663, 4.54056490638449807872541724447, 4.709911320456718461368277460600, 5.930042429938118670468957932259, 6.29743799237743774707086850219, 7.509113252599601085601776248738, 8.167041509004095876695409580527, 9.681629463830000614092938933495, 10.707684981532854471852230215148, 11.17694913836739447958276737682, 12.04446472889535527033059452355, 12.64395635828013284291406113344, 13.28903652696543844169208316321, 14.37170232631767521610642130095, 15.09651356380517901409550666375, 15.75424323013539604636745118904, 16.79037434638420328719084926021, 17.4906364116616035214537605970, 18.29125239293497508730423751707, 19.192459253465080843389681067992, 19.9176060294934097351456395210, 20.99044424256090626195384034998, 21.532824911874657510168559783300

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