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Properties

Label 1-569-569.46-r1-0-0
Degree $1$
Conductor $569$
Sign $-0.611 + 0.791i$
Analytic cond. $61.1475$
Root an. cond. $61.1475$
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  + (0.346 + 0.937i)2-s + (0.0331 − 0.999i)3-s + (−0.759 + 0.650i)4-s + (−0.921 − 0.387i)5-s + (0.948 − 0.315i)6-s + (−0.467 − 0.883i)7-s + (−0.873 − 0.487i)8-s + (−0.997 − 0.0663i)9-s + (0.0442 − 0.999i)10-s + (−0.496 − 0.867i)11-s + (0.624 + 0.780i)12-s + (−0.304 − 0.952i)13-s + (0.666 − 0.745i)14-s + (−0.418 + 0.908i)15-s + (0.154 − 0.988i)16-s + (−0.544 − 0.839i)17-s + ⋯
L(s)  = 1  + (0.346 + 0.937i)2-s + (0.0331 − 0.999i)3-s + (−0.759 + 0.650i)4-s + (−0.921 − 0.387i)5-s + (0.948 − 0.315i)6-s + (−0.467 − 0.883i)7-s + (−0.873 − 0.487i)8-s + (−0.997 − 0.0663i)9-s + (0.0442 − 0.999i)10-s + (−0.496 − 0.867i)11-s + (0.624 + 0.780i)12-s + (−0.304 − 0.952i)13-s + (0.666 − 0.745i)14-s + (−0.418 + 0.908i)15-s + (0.154 − 0.988i)16-s + (−0.544 − 0.839i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.611 + 0.791i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ -0.611 + 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2447191308 - 0.4983942106i\)
\(L(\frac12)\) \(\approx\) \(-0.2447191308 - 0.4983942106i\)
\(L(1)\) \(\approx\) \(0.6529674645 - 0.2303856981i\)
\(L(1)\) \(\approx\) \(0.6529674645 - 0.2303856981i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.346 + 0.937i)T \)
3 \( 1 + (0.0331 - 0.999i)T \)
5 \( 1 + (-0.921 - 0.387i)T \)
7 \( 1 + (-0.467 - 0.883i)T \)
11 \( 1 + (-0.496 - 0.867i)T \)
13 \( 1 + (-0.304 - 0.952i)T \)
17 \( 1 + (-0.544 - 0.839i)T \)
19 \( 1 + (-0.997 + 0.0773i)T \)
23 \( 1 + (0.438 - 0.898i)T \)
29 \( 1 + (-0.208 - 0.977i)T \)
31 \( 1 + (-0.973 + 0.230i)T \)
37 \( 1 + (0.955 + 0.294i)T \)
41 \( 1 + (-0.325 - 0.945i)T \)
43 \( 1 + (-0.937 - 0.346i)T \)
47 \( 1 + (0.878 - 0.477i)T \)
53 \( 1 + (0.315 + 0.948i)T \)
59 \( 1 + (-0.477 - 0.878i)T \)
61 \( 1 + (0.930 - 0.367i)T \)
67 \( 1 + (-0.544 + 0.839i)T \)
71 \( 1 + (0.873 - 0.487i)T \)
73 \( 1 + (-0.0773 - 0.997i)T \)
79 \( 1 + (-0.801 - 0.598i)T \)
83 \( 1 + (-0.982 + 0.186i)T \)
89 \( 1 + (0.230 - 0.973i)T \)
97 \( 1 + (0.0110 - 0.999i)T \)
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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

−23.33283005450798450210628159634, −22.56799169276296293992650891066, −21.6693538255458817938966323459, −21.45539413535049486916573117562, −20.14038877442846633081911325863, −19.66802858503515928620165487946, −18.86347012405552723805232609728, −17.99022836239810114772426238185, −16.779036330388243081321617539638, −15.71761375806981330370980758578, −14.90914291502185105219617103671, −14.7091194439835909813862647503, −13.13982859215525919454592571074, −12.37819242568749679542365107827, −11.45521866438295409534119176182, −10.88227812527146577025764776433, −9.88812665805054057191904469376, −9.13087728695492766974359877884, −8.31757979932092146806190297052, −6.77178504013702683848476083576, −5.57637722805208418120816988720, −4.5731904305676424008726131844, −3.90775957992062069763613994995, −2.91512045285965621628089380030, −2.019188555896270570119068542587, 0.237297267504371373378889173965, 0.51997619303773776305122430160, 2.74467313346721284521951126624, 3.67841664014945985337539787019, 4.78485782985044322693704958965, 5.8398709173024813704790333658, 6.8393219309811039247782144003, 7.520477988392332149829376772257, 8.23681424085270100541136596916, 8.98518536585681222122516943649, 10.56690538460409038289841117332, 11.615678845010733096992458126501, 12.71700041181723968273396487052, 13.09576977123307707338249792179, 13.91203854936456332249005163769, 14.93508160349992163355291043158, 15.78862833404368667046154416272, 16.7271067552322064730470663783, 17.17233802280718921119030930311, 18.39030524114377256389242730704, 18.97555856005864062028114272644, 20.00072186758775679687378619712, 20.65290908876881869456496126349, 22.125537182559657693191196081772, 22.92715763188082198992848922742

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