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Properties

Label 1-1045-1045.108-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.627 + 0.778i$
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  + (−0.694 − 0.719i)2-s + (−0.529 − 0.848i)3-s + (−0.0348 + 0.999i)4-s + (−0.241 + 0.970i)6-s + (−0.743 − 0.669i)7-s + (0.743 − 0.669i)8-s + (−0.438 + 0.898i)9-s + (0.866 − 0.5i)12-s + (0.0697 + 0.997i)13-s + (0.0348 + 0.999i)14-s + (−0.997 − 0.0697i)16-s + (0.898 − 0.438i)17-s + (0.951 − 0.309i)18-s + (−0.173 + 0.984i)21-s + (0.342 − 0.939i)23-s + (−0.961 − 0.275i)24-s + ⋯
L(s)  = 1  + (−0.694 − 0.719i)2-s + (−0.529 − 0.848i)3-s + (−0.0348 + 0.999i)4-s + (−0.241 + 0.970i)6-s + (−0.743 − 0.669i)7-s + (0.743 − 0.669i)8-s + (−0.438 + 0.898i)9-s + (0.866 − 0.5i)12-s + (0.0697 + 0.997i)13-s + (0.0348 + 0.999i)14-s + (−0.997 − 0.0697i)16-s + (0.898 − 0.438i)17-s + (0.951 − 0.309i)18-s + (−0.173 + 0.984i)21-s + (0.342 − 0.939i)23-s + (−0.961 − 0.275i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.627 + 0.778i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.627 + 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1087795256 - 0.2273812713i\)
\(L(\frac12)\) \(\approx\) \(-0.1087795256 - 0.2273812713i\)
\(L(1)\) \(\approx\) \(0.4046616146 - 0.3039530989i\)
\(L(1)\) \(\approx\) \(0.4046616146 - 0.3039530989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.694 - 0.719i)T \)
3 \( 1 + (-0.529 - 0.848i)T \)
7 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (0.0697 + 0.997i)T \)
17 \( 1 + (0.898 - 0.438i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (-0.882 + 0.469i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.848 + 0.529i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (0.139 - 0.990i)T \)
53 \( 1 + (0.829 - 0.559i)T \)
59 \( 1 + (0.990 - 0.139i)T \)
61 \( 1 + (0.961 - 0.275i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (-0.559 + 0.829i)T \)
73 \( 1 + (-0.927 - 0.374i)T \)
79 \( 1 + (-0.241 - 0.970i)T \)
83 \( 1 + (-0.406 + 0.913i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (-0.694 - 0.719i)T \)
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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

−22.22696403484436892601727981392, −21.295800610389747544582659753484, −20.456280322768017423393014962653, −19.50878452240291579581817334094, −18.87408183312293821464562973128, −17.87110751107651398188727999373, −17.35236528325931212890683530670, −16.46052141578754949792572371988, −15.89052588445230043857237538223, −15.195180470810452722306409732875, −14.68164080475574255731268465005, −13.45709442895423410312069366578, −12.42467125818403637321441149961, −11.52973365640059629830184944746, −10.51006570571055403984605781238, −10.009920753143140171666075686, −9.21477566475518059989790471512, −8.502312635938084525782825207931, −7.44939352673854902672129457019, −6.43173203365399579167787228172, −5.57781978528593422661511093037, −5.265570347567318481797733760156, −3.8292320563338687975840186358, −2.88124266803385768754299996203, −1.27828703302175854086959062375, 0.16575549537594454259357640684, 1.25036857348148340081812309457, 2.20946268668710652511918850198, 3.26738658588931089618568572015, 4.23941038955035362985016238233, 5.46301453174049162418241027576, 6.83238999700760379591442272180, 7.020632402134166833921136239522, 8.1068560955570381189186641054, 8.988359090534425550730732230348, 9.98766291895363472478603893390, 10.60715855380443541726485676, 11.63245067201623692947911356739, 12.05090266417784745820799439500, 13.09742704779967074847219307799, 13.48052568816794539889938086219, 14.51700785703474873235849928099, 16.0871012718096589142885556237, 16.710850909681347975115881943, 17.05717028828107776354177287654, 18.14661782862902013020313930630, 18.92709897867024548047273719941, 19.10304760659195276531955530443, 20.22367633669488564031614762156, 20.74647547017506672370726244472

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