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Properties

Label 1-1045-1045.289-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.0627 - 0.998i$
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  + (−0.848 + 0.529i)2-s + (−0.559 + 0.829i)3-s + (0.438 − 0.898i)4-s + (0.0348 − 0.999i)6-s + (0.104 − 0.994i)7-s + (0.104 + 0.994i)8-s + (−0.374 − 0.927i)9-s + (0.5 + 0.866i)12-s + (0.615 − 0.788i)13-s + (0.438 + 0.898i)14-s + (−0.615 − 0.788i)16-s + (0.374 − 0.927i)17-s + (0.809 + 0.587i)18-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (−0.882 − 0.469i)24-s + ⋯
L(s)  = 1  + (−0.848 + 0.529i)2-s + (−0.559 + 0.829i)3-s + (0.438 − 0.898i)4-s + (0.0348 − 0.999i)6-s + (0.104 − 0.994i)7-s + (0.104 + 0.994i)8-s + (−0.374 − 0.927i)9-s + (0.5 + 0.866i)12-s + (0.615 − 0.788i)13-s + (0.438 + 0.898i)14-s + (−0.615 − 0.788i)16-s + (0.374 − 0.927i)17-s + (0.809 + 0.587i)18-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (−0.882 − 0.469i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.0627 - 0.998i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.0627 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2939058368 - 0.3129512976i\)
\(L(\frac12)\) \(\approx\) \(0.2939058368 - 0.3129512976i\)
\(L(1)\) \(\approx\) \(0.5474260180 + 0.05317966186i\)
\(L(1)\) \(\approx\) \(0.5474260180 + 0.05317966186i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.848 + 0.529i)T \)
3 \( 1 + (-0.559 + 0.829i)T \)
7 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (0.615 - 0.788i)T \)
17 \( 1 + (0.374 - 0.927i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.997 + 0.0697i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.559 - 0.829i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (0.241 - 0.970i)T \)
53 \( 1 + (-0.990 - 0.139i)T \)
59 \( 1 + (-0.241 - 0.970i)T \)
61 \( 1 + (-0.882 + 0.469i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.990 - 0.139i)T \)
73 \( 1 + (-0.961 - 0.275i)T \)
79 \( 1 + (0.0348 + 0.999i)T \)
83 \( 1 + (-0.669 + 0.743i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (-0.848 + 0.529i)T \)
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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

−21.7036900606716347220492761410, −20.999928806717730494394451517476, −19.90225158962673058339407917344, −19.18312402410095154129282650636, −18.59802703878287174943616537794, −18.08179308723070961167818027573, −17.23609452686305726545686704011, −16.545599633655492704554698173997, −15.77356433553774309479741452307, −14.7151479950860168113391069289, −13.52408326515931187841398937477, −12.7354767221319263563568520351, −12.138548168683097307918343593908, −11.29658238808562830272225330860, −10.890840213576284519030175050935, −9.58283587355244095648547466099, −8.90468687559916768550932583704, −7.99600566792438313249736540418, −7.345991442334447542711090125387, −6.21913726173915479476356174590, −5.67934389713177555532428060398, −4.21116823969963721007736734845, −3.04825701922994561997093388941, −1.9190736644988774002751573542, −1.44364826224334381692312757422, 0.27961943968009636837534762873, 1.29059743603572451284637605773, 2.91950236144795407992158869885, 4.034163394807256685536640356053, 5.02076903565920364888897507569, 5.7929120782359593457211532637, 6.69729657424196867922070317961, 7.55428499891144646370276514922, 8.44088002607368608147348568002, 9.43552535500723980051629052937, 10.04337456972162015087570116882, 10.90202926366041341644502925957, 11.22943098617858461835494408949, 12.492562710053735516929230320249, 13.696968800608647064327500531811, 14.56047374964029018484178252002, 15.216237361851389116493930819693, 16.28225953246274237829869865927, 16.452381966857640247553336789121, 17.36370062452371543049287943460, 18.05338380131287659223943954780, 18.711101108448150277055940995385, 20.080759277842057504143074007117, 20.34723829669489536394246713451, 21.082769656256441780546188378681

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