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Properties

Label 1-1045-1045.43-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.492 + 0.870i$
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  + (0.642 + 0.766i)2-s + (0.342 − 0.939i)3-s + (−0.173 + 0.984i)4-s + (0.939 − 0.342i)6-s + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)8-s + (−0.766 − 0.642i)9-s + (0.866 + 0.5i)12-s + (0.342 + 0.939i)13-s + (−0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + (0.642 + 0.766i)17-s − i·18-s + (−0.766 + 0.642i)21-s + (0.984 + 0.173i)23-s + (0.173 + 0.984i)24-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (0.342 − 0.939i)3-s + (−0.173 + 0.984i)4-s + (0.939 − 0.342i)6-s + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)8-s + (−0.766 − 0.642i)9-s + (0.866 + 0.5i)12-s + (0.342 + 0.939i)13-s + (−0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + (0.642 + 0.766i)17-s − i·18-s + (−0.766 + 0.642i)21-s + (0.984 + 0.173i)23-s + (0.173 + 0.984i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.492 + 0.870i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.492 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.691347541 + 0.9866716300i\)
\(L(\frac12)\) \(\approx\) \(1.691347541 + 0.9866716300i\)
\(L(1)\) \(\approx\) \(1.387372418 + 0.3793196450i\)
\(L(1)\) \(\approx\) \(1.387372418 + 0.3793196450i\)

Euler product

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

−21.36835861954082406866325708202, −20.66473044452120971230156808014, −20.147751691630816195388975271669, −19.18988498456688150628401961153, −18.72652014429859147144979245754, −17.55805782852232471224931789349, −16.42885812218166173919615501911, −15.65151716075859530037742693258, −15.16925186365747124416991449882, −14.28813091807459120791001523482, −13.47308896450474200614962153391, −12.73937035477200278674771256347, −11.866707653165038336291386574368, −10.96818550378384161380122818751, −10.252905371172795481647330597182, −9.50420675369281771144581826245, −8.94385917147510838870341109354, −7.743991777698282954353749934803, −6.30098792606573315450460471309, −5.57101229895198791349594418374, −4.795229769263966366180758876769, −3.76535704615003479631360654237, −3.004064965171087537415239786413, −2.45961669161654975866278230180, −0.73353970400008763382281476763, 1.12129991381499162938468280314, 2.51270094789803504526533590317, 3.42462432613631191429126303599, 4.147566608719727680395445170928, 5.51544085621727877948852824058, 6.27780470402828592964505730512, 7.03447965159904834994277708807, 7.5197357660745125607481241678, 8.70769657070848433324067583134, 9.16732946054220176657605869887, 10.581014198656381768136659367633, 11.68361107949399356738088443591, 12.60002556983836210591070249304, 12.940425980866264872020287292519, 13.956404589319585860917907550710, 14.294572641829494161996504507982, 15.30127552037539129141293165719, 16.27274826131425106870942947273, 16.85035619684475550643079246219, 17.659283841287121336480700014099, 18.54129532207216871802735642693, 19.33568790914555887914092525585, 20.01002289724944531866609492252, 21.098832765763547805385584257240, 21.6688096257902937693858153452

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