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Properties

Label 1-1045-1045.239-r1-0-0
Degree $1$
Conductor $1045$
Sign $0.472 - 0.881i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.472 - 0.881i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ 0.472 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6387064333 - 0.3824125877i\)
\(L(\frac12)\) \(\approx\) \(0.6387064333 - 0.3824125877i\)
\(L(1)\) \(\approx\) \(0.5192607058 - 0.07606390495i\)
\(L(1)\) \(\approx\) \(0.5192607058 - 0.07606390495i\)

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

−21.538270351822553787998152126558, −20.70532004221082819124585946359, −19.82229973988925812699115598372, −18.94309191841332896497615303956, −18.367485354396205092181248327971, −17.568207858829102871957334939934, −16.87063968715732081390340594446, −16.148590233704711805402537423174, −15.51121372468986234722743505325, −14.88812220105465305055258940911, −13.11953368462511409427714337735, −12.603701033964793830388421234556, −11.70972658526789621110213143116, −11.03568092448537754333446559000, −10.16945076562095393801825042892, −9.6350460945586837780392412432, −8.75078285002419287834974625012, −7.794276236818686232751947928744, −6.72330203318546486711271469022, −6.020405985893233473539353472461, −5.31385030355495348027864170315, −3.72148890140224948721801204682, −3.1254996344568305816101518679, −1.67176771002240594829028930084, −0.64701709831477695575260931485, 0.435677866519163037233660549789, 1.188174271422989629208770606646, 2.354541786305061892576841604927, 3.63062737766885053195844613374, 4.933261655499909314591550700118, 5.97775467910433264500331684053, 6.65264483643817094707044888865, 7.24183210911659357777391846411, 8.1508814854117880022176523338, 9.29531630247385401120759490863, 9.964988089158229087496749092969, 10.81540569185010667778338016947, 11.45239048773067442568727192915, 12.28500524397781999204239315279, 13.21109870319694454473219996053, 14.07535959151505891219539606953, 15.262096959552636443902346112113, 16.17467571696589666856444671945, 16.673375318378063036389049140561, 17.10668914812111316073971348069, 18.27387299501880571108966257282, 18.70516205662028811089037986890, 19.295633416327068148296283782295, 20.38096026427498371284838088421, 20.95953687357726424646161612587

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