Properties

Label 1-1045-1045.747-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.0148 + 0.999i$
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  + (−0.984 − 0.173i)2-s + (0.642 − 0.766i)3-s + (0.939 + 0.342i)4-s + (−0.766 + 0.642i)6-s + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)8-s + (−0.173 − 0.984i)9-s + (0.866 − 0.5i)12-s + (0.642 + 0.766i)13-s + (0.939 − 0.342i)14-s + (0.766 + 0.642i)16-s + (−0.984 − 0.173i)17-s + i·18-s + (−0.173 + 0.984i)21-s + (−0.342 + 0.939i)23-s + (−0.939 + 0.342i)24-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (0.642 − 0.766i)3-s + (0.939 + 0.342i)4-s + (−0.766 + 0.642i)6-s + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)8-s + (−0.173 − 0.984i)9-s + (0.866 − 0.5i)12-s + (0.642 + 0.766i)13-s + (0.939 − 0.342i)14-s + (0.766 + 0.642i)16-s + (−0.984 − 0.173i)17-s + i·18-s + (−0.173 + 0.984i)21-s + (−0.342 + 0.939i)23-s + (−0.939 + 0.342i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0148 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0148 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.0148 + 0.999i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (747, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.0148 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3026825923 + 0.3072027372i\)
\(L(\frac12)\) \(\approx\) \(0.3026825923 + 0.3072027372i\)
\(L(1)\) \(\approx\) \(0.6421976249 - 0.08425023052i\)
\(L(1)\) \(\approx\) \(0.6421976249 - 0.08425023052i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.984 - 0.173i)T \)
3 \( 1 + (0.642 - 0.766i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (-0.984 + 0.173i)T \)
53 \( 1 + (-0.342 + 0.939i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.642 + 0.766i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.984 + 0.173i)T \)
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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

−21.02127041597851823776364226138, −20.3812359520981834287793921508, −19.74000879727630318378849021958, −19.249955658998977060378686847389, −18.18519501574042258692577847113, −17.46078713509994527668151011146, −16.35833748011061370192160510598, −16.1187680507647584477811026123, −15.270617070598025110464753127679, −14.54464145703096109567796367109, −13.50057794274579824890509973055, −12.73514728449359224964986036995, −11.37862031537337788705222138111, −10.611824628147152759465843345144, −10.07395656458298568447237620495, −9.28704739923670396412707611161, −8.498862520078882541722166164799, −7.85055037197312509609758474757, −6.75122914213044449935969246167, −6.01978800506258351046353298712, −4.77059956371544664326168394467, −3.614433344866999944150685436657, −2.87657244200428769228164626778, −1.80392673999584444033590749553, −0.22124769803328617619746999878, 1.34264541940747586818868203039, 2.17004746165161684463573856637, 3.0670664428473385453752127826, 3.89540486650991995473082355313, 5.74109402520002946014833670503, 6.615751765028287528315390558236, 7.09460623511647646539529710098, 8.17285950545465702567835735114, 8.969433991976889514879154819769, 9.33964857555419332310161315070, 10.38546705445730418384268125617, 11.54351410787359236511960697997, 12.03448499303975626831013737829, 13.07376447369569354036084345562, 13.60090814278387756890500956083, 14.8562467240185187143508965543, 15.59259989028158415463075968475, 16.28798939699968109103108802982, 17.25795640614505080775784437870, 18.125429753741119520460628914902, 18.678591554142348790327892599231, 19.27281018892976362389541439947, 20.00760679983967315844715325542, 20.58810017192378981622843943161, 21.58754203025158478750606628031

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