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Properties

Label 1-283-283.245-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.809 + 0.587i$
Analytic cond. $30.4125$
Root an. cond. $30.4125$
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  + (0.645 − 0.763i)2-s + (0.296 − 0.955i)3-s + (−0.166 − 0.986i)4-s + (0.824 − 0.565i)5-s + (−0.538 − 0.842i)6-s + (0.359 − 0.933i)7-s + (−0.860 − 0.509i)8-s + (−0.824 − 0.565i)9-s + (0.100 − 0.994i)10-s + (−0.166 + 0.986i)11-s + (−0.991 − 0.133i)12-s + (−0.538 − 0.842i)13-s + (−0.480 − 0.876i)14-s + (−0.296 − 0.955i)15-s + (−0.944 + 0.328i)16-s + (−0.480 + 0.876i)17-s + ⋯
L(s)  = 1  + (0.645 − 0.763i)2-s + (0.296 − 0.955i)3-s + (−0.166 − 0.986i)4-s + (0.824 − 0.565i)5-s + (−0.538 − 0.842i)6-s + (0.359 − 0.933i)7-s + (−0.860 − 0.509i)8-s + (−0.824 − 0.565i)9-s + (0.100 − 0.994i)10-s + (−0.166 + 0.986i)11-s + (−0.991 − 0.133i)12-s + (−0.538 − 0.842i)13-s + (−0.480 − 0.876i)14-s + (−0.296 − 0.955i)15-s + (−0.944 + 0.328i)16-s + (−0.480 + 0.876i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.809 + 0.587i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.809 + 0.587i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9047545352 - 2.786532069i\)
\(L(\frac12)\) \(\approx\) \(-0.9047545352 - 2.786532069i\)
\(L(1)\) \(\approx\) \(0.8083027965 - 1.512465187i\)
\(L(1)\) \(\approx\) \(0.8083027965 - 1.512465187i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.645 - 0.763i)T \)
3 \( 1 + (0.296 - 0.955i)T \)
5 \( 1 + (0.824 - 0.565i)T \)
7 \( 1 + (0.359 - 0.933i)T \)
11 \( 1 + (-0.166 + 0.986i)T \)
13 \( 1 + (-0.538 - 0.842i)T \)
17 \( 1 + (-0.480 + 0.876i)T \)
19 \( 1 + (0.538 + 0.842i)T \)
23 \( 1 + (0.231 - 0.972i)T \)
29 \( 1 + (0.964 + 0.264i)T \)
31 \( 1 + (-0.695 - 0.718i)T \)
37 \( 1 + (0.892 + 0.451i)T \)
41 \( 1 + (0.860 - 0.509i)T \)
43 \( 1 + (-0.593 - 0.805i)T \)
47 \( 1 + (0.538 - 0.842i)T \)
53 \( 1 + (0.944 + 0.328i)T \)
59 \( 1 + (-0.741 - 0.670i)T \)
61 \( 1 + (0.100 + 0.994i)T \)
67 \( 1 + (-0.991 + 0.133i)T \)
71 \( 1 + (-0.166 + 0.986i)T \)
73 \( 1 + (0.695 - 0.718i)T \)
79 \( 1 + (-0.593 + 0.805i)T \)
83 \( 1 + (0.784 - 0.619i)T \)
89 \( 1 + (0.593 + 0.805i)T \)
97 \( 1 + (0.231 - 0.972i)T \)
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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

−25.814603143193395910627725463571, −25.01245996558428740952837204242, −24.326650206115653817789184909106, −23.03412879740858622137903370975, −21.92407355279830802419574034005, −21.67479360775410275780275565253, −21.082260969311709382446076187401, −19.63587549778120655397092665919, −18.29949581016524792574383968205, −17.524524692080100965621544303273, −16.38649495334082448531748630190, −15.70569234889222618187986364797, −14.74654027502415342320829442562, −14.06999391113896050121099842135, −13.356141592885655397731925943944, −11.73266250919673665277581429040, −11.04626245082102893475157529333, −9.37594373471313302246070990149, −9.01130651716431214203557952532, −7.65145533747126943797819356770, −6.35271293213334202481582780547, −5.41258913831476314276164348769, −4.67685761217214243184558016395, −3.13064599243095724266772807188, −2.47309018502932998497523010470, 0.66084592776325349515700002053, 1.67338424883703932327961158294, 2.57384123203848041467932410988, 4.089038231612152244963055921744, 5.20352142304235635480021564801, 6.25741006941771386240479235147, 7.4170786955938749768600237680, 8.637124052604018686775844567379, 9.91222196526306130748751720105, 10.606949477131472313213939802913, 12.0801603124770149994525092848, 12.75809909603034084100805374889, 13.40615111676075151072408415116, 14.31664155799733168389236522952, 15.04019195689736657399636520618, 16.84454396366404708874698645526, 17.72443992108614996065554976749, 18.395510505556773141463980152577, 19.7962087833840682819447518127, 20.2560065597163794269548432776, 20.87659945293387872041319322695, 22.15585715922935162292650694025, 23.06216983284833808564658313214, 23.868508838942082983310069992067, 24.64755274158589648718819839496

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