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Properties

Label 1-1045-1045.354-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.392 + 0.919i$
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  + (−0.913 + 0.406i)2-s + (0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (−0.913 − 0.406i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + 12-s + (0.104 − 0.994i)13-s + (−0.669 − 0.743i)14-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (−0.309 − 0.951i)18-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (−0.913 + 0.406i)24-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (−0.913 − 0.406i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + 12-s + (0.104 − 0.994i)13-s + (−0.669 − 0.743i)14-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (−0.309 − 0.951i)18-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (−0.913 + 0.406i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.392 + 0.919i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (354, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.392 + 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7012168593 + 1.062069090i\)
\(L(\frac12)\) \(\approx\) \(0.7012168593 + 1.062069090i\)
\(L(1)\) \(\approx\) \(0.8115533159 + 0.4905264319i\)
\(L(1)\) \(\approx\) \(0.8115533159 + 0.4905264319i\)

Euler product

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

−20.911643227081516409333021927373, −20.44529534774969080054348764411, −19.49557019467384634811430279402, −19.16926081525623644823279419184, −18.27669377906299068000235902155, −17.50592860567168392363861972186, −16.90727244644199081881437504234, −16.02905664669587857584450415673, −14.90867362479265876683989911407, −14.15215875251346773723710926890, −13.28369913023788561279359190212, −12.53013113325061320767507526269, −11.71022057049618417423650862599, −10.80466605568385786961669747992, −10.0849369536476748356604649070, −8.9880447376716245812046975204, −8.499345362837008880766083593325, −7.56401876673254894847822959341, −6.942574916909427162088496187609, −6.19406459121135731964674924182, −4.341559691360704819446339255119, −3.62487289823886044344500385219, −2.475327870862908782108073070302, −1.66176142513184237025672186072, −0.73938684937716177818304589520, 1.22746575912498939919536550797, 2.5613873948244552028889892537, 3.01823416029841913807657484882, 4.67756339217391788602690219486, 5.35688712657833724635404242580, 6.28423571023794242767790799509, 7.573160487054793799793072019243, 8.146931163061098533153069955262, 8.934817518604480869740474309938, 9.58791691845756433390982629271, 10.32527004119811576435387948259, 11.23675419482234069726255770997, 11.96495663480894099869204239880, 13.32911857999329200024961353319, 14.19805437814896840200930632028, 15.1007031867560786798968892615, 15.50725481381241775355334226743, 16.11684648689927001514802017090, 17.13866295522290566084566653806, 17.91453010049827050863117630542, 18.645765010015731621973292186177, 19.45809347318028055736119740510, 20.121834227076657364541293394911, 20.908168331523372020841888522820, 21.512362879558094773419130603658

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