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Properties

Label 1-1045-1045.84-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.983 - 0.180i$
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  + (−0.669 + 0.743i)2-s + (−0.104 + 0.994i)3-s + (−0.104 − 0.994i)4-s + (−0.669 − 0.743i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + 12-s + (0.978 + 0.207i)13-s + (0.104 − 0.994i)14-s + (−0.978 + 0.207i)16-s + (−0.978 + 0.207i)17-s + (0.809 − 0.587i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.669 + 0.743i)24-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.104 + 0.994i)3-s + (−0.104 − 0.994i)4-s + (−0.669 − 0.743i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + 12-s + (0.978 + 0.207i)13-s + (0.104 − 0.994i)14-s + (−0.978 + 0.207i)16-s + (−0.978 + 0.207i)17-s + (0.809 − 0.587i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.669 + 0.743i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.983 - 0.180i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.983 - 0.180i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3985990447 - 0.03618824444i\)
\(L(\frac12)\) \(\approx\) \(0.3985990447 - 0.03618824444i\)
\(L(1)\) \(\approx\) \(0.4906698317 + 0.2919935537i\)
\(L(1)\) \(\approx\) \(0.4906698317 + 0.2919935537i\)

Euler product

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

−21.41799142248316801629738285341, −20.49732912937454485188851315340, −19.707170418616998497213822178630, −19.44930425473285953431534685372, −18.359135556560785093937264441619, −17.957057697507439184747245550605, −17.10700596438340365022029891297, −16.36537798827695373411017445529, −15.55398536763060397062467183503, −13.99706292522771043936608297819, −13.4249481294365841169137022285, −12.800948816095602079187696190932, −12.089183668569665294960269752117, −10.980910546964563845955813099863, −10.712839816549388908082203988455, −9.366647114760390088010253335371, −8.78879042673879662649584171733, −7.821276404581102001893827729880, −6.992141324983347070368888677106, −6.41858825151054514900542923991, −5.07228921552452584171754920256, −3.64768264747100379619877553293, −3.07764638837829552190737101860, −1.861377224004576575451878275, −1.0284271789183483193270888093, 0.247345825221865357496547287603, 1.980436864914258657708785583054, 3.18856116863509545506346115782, 4.295947338548753589543626006073, 5.1523006544208700822387081130, 6.27392553074017518713351737180, 6.45058419763073496860919611518, 8.0052131136014936318100775931, 8.78389016379229569526947086185, 9.326170585601352287848943436527, 10.11205989680057360286514313662, 10.93577224098828534109946717814, 11.633840011538535036740202381334, 13.011089697415712076435046796628, 13.81978161616853268671851035233, 14.87358652582372017537118756294, 15.454301581143038738906125462593, 15.98323750732031457512438520271, 16.73492604854877131008266102003, 17.391501758872894738013696479864, 18.4264961338266797595044832792, 18.99231360478306142404678857315, 19.957881462859036959711332856548, 20.61573022325113126839589870876, 21.58776908415043637003085223867

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