↑

Properties

Label 1-569-569.221-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.996 + 0.0807i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.937 + 0.346i)2-s + (0.730 + 0.683i)3-s + (0.759 + 0.650i)4-s + (−0.921 + 0.387i)5-s + (0.448 + 0.894i)6-s + (−0.883 − 0.467i)7-s + (0.487 + 0.873i)8-s + (0.0663 + 0.997i)9-s + (−0.999 + 0.0442i)10-s + (−0.964 + 0.262i)11-s + (0.110 + 0.993i)12-s + (−0.952 − 0.304i)13-s + (−0.666 − 0.745i)14-s + (−0.937 − 0.346i)15-s + (0.154 + 0.988i)16-s + (−0.839 − 0.544i)17-s + ⋯
L(s)  = 1  + (0.937 + 0.346i)2-s + (0.730 + 0.683i)3-s + (0.759 + 0.650i)4-s + (−0.921 + 0.387i)5-s + (0.448 + 0.894i)6-s + (−0.883 − 0.467i)7-s + (0.487 + 0.873i)8-s + (0.0663 + 0.997i)9-s + (−0.999 + 0.0442i)10-s + (−0.964 + 0.262i)11-s + (0.110 + 0.993i)12-s + (−0.952 − 0.304i)13-s + (−0.666 − 0.745i)14-s + (−0.937 − 0.346i)15-s + (0.154 + 0.988i)16-s + (−0.839 − 0.544i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.996 + 0.0807i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.996 + 0.0807i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06312580571 + 1.560611518i\)
\(L(\frac12)\) \(\approx\) \(0.06312580571 + 1.560611518i\)
\(L(1)\) \(\approx\) \(1.115758203 + 0.9359479115i\)
\(L(1)\) \(\approx\) \(1.115758203 + 0.9359479115i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.937 + 0.346i)T \)
3 \( 1 + (0.730 + 0.683i)T \)
5 \( 1 + (-0.921 + 0.387i)T \)
7 \( 1 + (-0.883 - 0.467i)T \)
11 \( 1 + (-0.964 + 0.262i)T \)
13 \( 1 + (-0.952 - 0.304i)T \)
17 \( 1 + (-0.839 - 0.544i)T \)
19 \( 1 + (-0.759 + 0.650i)T \)
23 \( 1 + (-0.325 + 0.945i)T \)
29 \( 1 + (0.839 - 0.544i)T \)
31 \( 1 + (0.525 + 0.850i)T \)
37 \( 1 + (0.883 + 0.467i)T \)
41 \( 1 + (0.325 - 0.945i)T \)
43 \( 1 + (0.937 - 0.346i)T \)
47 \( 1 + (0.283 + 0.958i)T \)
53 \( 1 + (0.448 + 0.894i)T \)
59 \( 1 + (0.283 + 0.958i)T \)
61 \( 1 + (-0.367 + 0.930i)T \)
67 \( 1 + (-0.839 + 0.544i)T \)
71 \( 1 + (0.487 - 0.873i)T \)
73 \( 1 + (-0.759 + 0.650i)T \)
79 \( 1 + (-0.598 - 0.801i)T \)
83 \( 1 + (-0.562 - 0.826i)T \)
89 \( 1 + (0.525 - 0.850i)T \)
97 \( 1 + (-0.699 + 0.714i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.01188850228854480890355549052, −21.9932452924181889682538833072, −21.20970444286923870816959163716, −20.18564797143057109846301050907, −19.580156458932948845304385457121, −19.14923779073008355292889110575, −18.22293642074157643378114283898, −16.691313434270906709897188555099, −15.719423872793700960540119259299, −15.1952577566666529299268900066, −14.35720264025920643522209452578, −13.104857659216661979906920806681, −12.839025460128321185627530758508, −12.09487765529523634168313421518, −11.11311698331322201410071361467, −9.94495104369808569936875295850, −8.85697321885937592770816995372, −7.932224842374216994305006905872, −6.901515740077474182850704737501, −6.18277426042238552182228993512, −4.79843717247931560407086226906, −3.94980463418337693095334493781, −2.77375910434306431411960585769, −2.2962263745306043128618160210, −0.48313025613431215271796840015, 2.52349902771705688569450366193, 2.99851272028017143021728664721, 4.13423116373520525156793824043, 4.628882991466919181723190502839, 5.95115487970637413021955967177, 7.2896918814148610699434615770, 7.617284588265013183967712090454, 8.750758772707876362790009921702, 10.14739234631501764977318559069, 10.66702523331944680748209337055, 11.88980684884369498973521824759, 12.78755415992589978833936916273, 13.62026217350055786635677269632, 14.43189162747758295080669108950, 15.421364572126168623143379227235, 15.68798864416458376684353137536, 16.47815830392741572575232827704, 17.57956879399672826231159001505, 19.11325238451582912334426396330, 19.71636026314810359902219251062, 20.36719862941180484538863688067, 21.25975313490833478824565028961, 22.16613408962468270255659376095, 22.79500952846613733160890049015, 23.46518779423405240465090850289

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