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Properties

Label 1-1045-1045.499-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: $\chi_{1045} (499, \cdot )$
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

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

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