Properties

Label 1-1045-1045.778-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.697 + 0.716i$
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  + (−0.951 − 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.309 − 0.951i)6-s + (0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + i·12-s + (0.951 + 0.309i)13-s + (−0.809 + 0.587i)14-s + (0.309 + 0.951i)16-s + (−0.951 + 0.309i)17-s + (0.587 − 0.809i)18-s + 21-s + i·23-s + (0.309 − 0.951i)24-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.309 − 0.951i)6-s + (0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + i·12-s + (0.951 + 0.309i)13-s + (−0.809 + 0.587i)14-s + (0.309 + 0.951i)16-s + (−0.951 + 0.309i)17-s + (0.587 − 0.809i)18-s + 21-s + i·23-s + (0.309 − 0.951i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.697 + 0.716i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (778, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.697 + 0.716i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4665607754 + 1.106294755i\)
\(L(\frac12)\) \(\approx\) \(0.4665607754 + 1.106294755i\)
\(L(1)\) \(\approx\) \(0.8379478104 + 0.2263493570i\)
\(L(1)\) \(\approx\) \(0.8379478104 + 0.2263493570i\)

Euler product

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

−20.53114461693566721721348196772, −20.36033568354487086050063758176, −19.23671152042089679283215110700, −18.59615434599390996704664190267, −18.09245812755847348284472639019, −17.48525618048224234017548058125, −16.434666874620852382552318950458, −15.38771778472045485609105076277, −15.047617326533891839139414919851, −14.04269660307652831088137211151, −13.23553300611962414692192086625, −12.139998856686979283127066367264, −11.4924833738438233628220096484, −10.636819403526392118086435802095, −9.469117206658132317142049075719, −8.69923437966162960718565627166, −8.27333594747943293035196717389, −7.44101738065567616293506881579, −6.39150979540211988543129385689, −5.92909191672428846933998133493, −4.57762966562081567939566501458, −3.002053773255699597039167836483, −2.25575576764207259333153952378, −1.385725169716751301803590771522, −0.31453894120341662383782156379, 1.21099095405684491601926533773, 2.057585273000712864231889596731, 3.291416393463167357789714583933, 3.94416721536674672978431294090, 4.93801157594523852220393902055, 6.32527063827491661869865596261, 7.31563907161715019570579156014, 8.138953811499999849754489074559, 8.81789471323485788664983500426, 9.5082504564782481723222556452, 10.59833110437069588511687642782, 10.8576084607542034849415141932, 11.71111903091122130270501384383, 13.0149081227062143808510366974, 13.81161338540411732530225658352, 14.64652599999486362357477513627, 15.72062941122545949492551869721, 16.05359750864801847142304419324, 17.09086107095362480926514014819, 17.64513959022930256776473175453, 18.56958730150156148376883748142, 19.525924383512884249084740588122, 20.10104571558803891232605801833, 20.636980018582316067856415125386, 21.49498723338291289285702807331

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