Properties

Label 1-569-569.71-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.317 - 0.948i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
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  + (0.110 + 0.993i)2-s + (0.346 − 0.937i)3-s + (−0.975 + 0.219i)4-s + (−0.448 − 0.894i)5-s + (0.970 + 0.240i)6-s + (0.999 + 0.0442i)7-s + (−0.325 − 0.945i)8-s + (−0.759 − 0.650i)9-s + (0.839 − 0.544i)10-s + (−0.304 − 0.952i)11-s + (−0.132 + 0.991i)12-s + (−0.633 − 0.773i)13-s + (0.0663 + 0.997i)14-s + (−0.993 + 0.110i)15-s + (0.903 − 0.428i)16-s + (0.367 + 0.930i)17-s + ⋯
L(s)  = 1  + (0.110 + 0.993i)2-s + (0.346 − 0.937i)3-s + (−0.975 + 0.219i)4-s + (−0.448 − 0.894i)5-s + (0.970 + 0.240i)6-s + (0.999 + 0.0442i)7-s + (−0.325 − 0.945i)8-s + (−0.759 − 0.650i)9-s + (0.839 − 0.544i)10-s + (−0.304 − 0.952i)11-s + (−0.132 + 0.991i)12-s + (−0.633 − 0.773i)13-s + (0.0663 + 0.997i)14-s + (−0.993 + 0.110i)15-s + (0.903 − 0.428i)16-s + (0.367 + 0.930i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.317 - 0.948i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.317 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5447290948 - 0.7571275533i\)
\(L(\frac12)\) \(\approx\) \(0.5447290948 - 0.7571275533i\)
\(L(1)\) \(\approx\) \(0.9280574197 - 0.1671732120i\)
\(L(1)\) \(\approx\) \(0.9280574197 - 0.1671732120i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.110 + 0.993i)T \)
3 \( 1 + (0.346 - 0.937i)T \)
5 \( 1 + (-0.448 - 0.894i)T \)
7 \( 1 + (0.999 + 0.0442i)T \)
11 \( 1 + (-0.304 - 0.952i)T \)
13 \( 1 + (-0.633 - 0.773i)T \)
17 \( 1 + (0.367 + 0.930i)T \)
19 \( 1 + (-0.219 + 0.975i)T \)
23 \( 1 + (-0.387 - 0.921i)T \)
29 \( 1 + (-0.930 - 0.367i)T \)
31 \( 1 + (0.616 - 0.787i)T \)
37 \( 1 + (0.0442 - 0.999i)T \)
41 \( 1 + (-0.921 + 0.387i)T \)
43 \( 1 + (-0.110 + 0.993i)T \)
47 \( 1 + (-0.826 + 0.562i)T \)
53 \( 1 + (-0.970 - 0.240i)T \)
59 \( 1 + (0.826 - 0.562i)T \)
61 \( 1 + (-0.984 - 0.176i)T \)
67 \( 1 + (0.367 - 0.930i)T \)
71 \( 1 + (-0.325 + 0.945i)T \)
73 \( 1 + (0.219 - 0.975i)T \)
79 \( 1 + (0.883 - 0.467i)T \)
83 \( 1 + (-0.0883 + 0.996i)T \)
89 \( 1 + (-0.616 - 0.787i)T \)
97 \( 1 + (-0.801 - 0.598i)T \)
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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

−23.294864664430590337686841790747, −22.336831933210966654181562635380, −21.80954199733842934130370338442, −21.00479763373273966843764536532, −20.27204459222493380575645763889, −19.58452007494285563233542467304, −18.65980938360621361310936067113, −17.816275066944340093623138942704, −17.01492686779277188724069801793, −15.56715193354629492661022433930, −14.92940416190607266894624113027, −14.23489202439297763465595152539, −13.53763284148560572217603773006, −11.90758223513925179678215082811, −11.55396342867297595101197025528, −10.611393537585494531198049741988, −9.91808447185406195362777386183, −9.08234647233160817569357139131, −7.99944738393686984496489089704, −7.07541983925013236327276326523, −5.1923776973909822691330799917, −4.68788130495162528440117750944, −3.72106120478964064278089764614, −2.716698142614195315324203160822, −1.89349756596425641003335379395, 0.43989740971541463740349907570, 1.69984560836873861513749535382, 3.32768185999460838255219793671, 4.42542799442757112362579452373, 5.51802451211286257359892937582, 6.14335801253982080911382538952, 7.61018396612153099008028156574, 8.15319547744482449916299803423, 8.423927342383812990278994593998, 9.720314720510166325425305622277, 11.194318103032556438696385204945, 12.35541470447330885488862854027, 12.800765168886556237732345560638, 13.76485354030168349674181547109, 14.62987662243584626446102350942, 15.197867953537359585781790321548, 16.47672293147763771190736249477, 17.04773963613990450916698363027, 17.879610272201221765403907717164, 18.74003639289282923329454021092, 19.44428082280813845491559474051, 20.57949550141006288300479037732, 21.25133451642293803394230801630, 22.50130703060884293245635013477, 23.44290255769419906485073973817

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