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Properties

Label 1-1045-1045.414-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.839 - 0.542i$
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  + (0.374 − 0.927i)2-s + (0.961 + 0.275i)3-s + (−0.719 − 0.694i)4-s + (0.615 − 0.788i)6-s + (0.913 + 0.406i)7-s + (−0.913 + 0.406i)8-s + (0.848 + 0.529i)9-s + (−0.5 − 0.866i)12-s + (−0.0348 + 0.999i)13-s + (0.719 − 0.694i)14-s + (0.0348 + 0.999i)16-s + (0.848 − 0.529i)17-s + (0.809 − 0.587i)18-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (−0.990 + 0.139i)24-s + ⋯
L(s)  = 1  + (0.374 − 0.927i)2-s + (0.961 + 0.275i)3-s + (−0.719 − 0.694i)4-s + (0.615 − 0.788i)6-s + (0.913 + 0.406i)7-s + (−0.913 + 0.406i)8-s + (0.848 + 0.529i)9-s + (−0.5 − 0.866i)12-s + (−0.0348 + 0.999i)13-s + (0.719 − 0.694i)14-s + (0.0348 + 0.999i)16-s + (0.848 − 0.529i)17-s + (0.809 − 0.587i)18-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (−0.990 + 0.139i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.839 - 0.542i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (414, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.839 - 0.542i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.600628534 - 0.7674487175i\)
\(L(\frac12)\) \(\approx\) \(2.600628534 - 0.7674487175i\)
\(L(1)\) \(\approx\) \(1.719636615 - 0.5295951706i\)
\(L(1)\) \(\approx\) \(1.719636615 - 0.5295951706i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.374 - 0.927i)T \)
3 \( 1 + (0.961 + 0.275i)T \)
7 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.0348 + 0.999i)T \)
17 \( 1 + (0.848 - 0.529i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.241 + 0.970i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.961 + 0.275i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (0.997 - 0.0697i)T \)
53 \( 1 + (-0.882 + 0.469i)T \)
59 \( 1 + (0.997 + 0.0697i)T \)
61 \( 1 + (-0.990 - 0.139i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.882 + 0.469i)T \)
73 \( 1 + (0.559 - 0.829i)T \)
79 \( 1 + (-0.615 - 0.788i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (-0.374 + 0.927i)T \)
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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

−21.424929866177571324543284877714, −21.005803023423810527546733114818, −20.16394332995330691892503738267, −19.223294104574105658163956290727, −18.35874340950362410001513916429, −17.60264058508384970385102648082, −17.034695830590693156754781178162, −15.825274135172913534027077015976, −15.233613729157347451433282778707, −14.4686520361654050020240009733, −14.000630231663325175245934128410, −13.08323100995119757580777093349, −12.51995981001829170673482197868, −11.40060004717219414910833528969, −10.1620751581672657816708751437, −9.33098178521030614349093135229, −8.33423253708116111557016104648, −7.74286965823117540706515835468, −7.329944117771100578654092698337, −6.02912595987916572438592542174, −5.251064364879598079109887317983, −4.09482989733424923007026449681, −3.524309157674976943712532859196, −2.33567659564681332738646074598, −1.03341008821315138395075549832, 1.34376211534323177524410056454, 2.097619645689723186889867736276, 2.96908056538782838965941362720, 3.92648136978928879507408612529, 4.73304478136013834650619297951, 5.44876577363971872835012972925, 6.8692587595450091956377895852, 7.97253324130039133223723188155, 8.8623711985017700703346664363, 9.3197830104009229546527870980, 10.38356858988835284345917680315, 11.031402345511394903955543636266, 12.07344355349089981730660632472, 12.60514868540983406233510285379, 13.86473595448286703936971065822, 14.21175375509656665794972338362, 14.810570032298589430466221876016, 15.750242645441582824164435234226, 16.74128360588606844756273903706, 17.971196938046097474805263983177, 18.69797811354898694599061961388, 19.10720519386374288067823590139, 20.293102473835414150675997804718, 20.55232255184448041702340114683, 21.48861881984708049088779369036

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