↑

Properties

Label 1-34e2-1156.55-r1-0-0
Degree $1$
Conductor $1156$
Sign $0.826 + 0.562i$
Analytic cond. $124.229$
Root an. cond. $124.229$
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  + (−0.895 + 0.445i)3-s + (0.526 + 0.850i)5-s + (0.798 + 0.602i)7-s + (0.602 − 0.798i)9-s + (0.361 − 0.932i)11-s + (−0.850 − 0.526i)13-s + (−0.850 − 0.526i)15-s + (−0.982 + 0.183i)19-s + (−0.982 − 0.183i)21-s + (0.798 + 0.602i)23-s + (−0.445 + 0.895i)25-s + (−0.183 + 0.982i)27-s + (0.361 − 0.932i)29-s + (0.526 − 0.850i)31-s + (0.0922 + 0.995i)33-s + ⋯
L(s)  = 1  + (−0.895 + 0.445i)3-s + (0.526 + 0.850i)5-s + (0.798 + 0.602i)7-s + (0.602 − 0.798i)9-s + (0.361 − 0.932i)11-s + (−0.850 − 0.526i)13-s + (−0.850 − 0.526i)15-s + (−0.982 + 0.183i)19-s + (−0.982 − 0.183i)21-s + (0.798 + 0.602i)23-s + (−0.445 + 0.895i)25-s + (−0.183 + 0.982i)27-s + (0.361 − 0.932i)29-s + (0.526 − 0.850i)31-s + (0.0922 + 0.995i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.826 + 0.562i$
Analytic conductor: \(124.229\)
Root analytic conductor: \(124.229\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1156,\ (1:\ ),\ 0.826 + 0.562i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.732063422 + 0.5337798963i\)
\(L(\frac12)\) \(\approx\) \(1.732063422 + 0.5337798963i\)
\(L(1)\) \(\approx\) \(0.9610874419 + 0.2310926056i\)
\(L(1)\) \(\approx\) \(0.9610874419 + 0.2310926056i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.895 + 0.445i)T \)
5 \( 1 + (0.526 + 0.850i)T \)
7 \( 1 + (0.798 + 0.602i)T \)
11 \( 1 + (0.361 - 0.932i)T \)
13 \( 1 + (-0.850 - 0.526i)T \)
19 \( 1 + (-0.982 + 0.183i)T \)
23 \( 1 + (0.798 + 0.602i)T \)
29 \( 1 + (0.361 - 0.932i)T \)
31 \( 1 + (0.526 - 0.850i)T \)
37 \( 1 + (-0.995 - 0.0922i)T \)
41 \( 1 + (-0.895 + 0.445i)T \)
43 \( 1 + (0.739 - 0.673i)T \)
47 \( 1 + (0.602 + 0.798i)T \)
53 \( 1 + (0.602 - 0.798i)T \)
59 \( 1 + (-0.273 + 0.961i)T \)
61 \( 1 + (0.961 - 0.273i)T \)
67 \( 1 + (0.982 - 0.183i)T \)
71 \( 1 + (-0.798 - 0.602i)T \)
73 \( 1 + (0.673 - 0.739i)T \)
79 \( 1 + (0.183 + 0.982i)T \)
83 \( 1 + (0.445 - 0.895i)T \)
89 \( 1 + (-0.850 + 0.526i)T \)
97 \( 1 + (0.798 + 0.602i)T \)
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

−21.13190917008900316140936513756, −20.25149771818828141520884781280, −19.5285430053118539313241056158, −18.56497370326122686534924545529, −17.56758410409099541797500894150, −17.23948616419117331759685193925, −16.77568717543839096458962026419, −15.76662174373716557202917131405, −14.64179861148389244649820279048, −13.95835149035463876315535126367, −12.99077119200198926538725603058, −12.364082623135160667669825254735, −11.78035091964879379106030353226, −10.67023766278300016895129989542, −10.14654737926206897575056640143, −9.03655030149656689537477821420, −8.21066743465780382305086407763, −7.02920975160022214199572390800, −6.72571810605435832115232245435, −5.34142705827839229322934712329, −4.803655733269827706052354838222, −4.210520986724715801618932337817, −2.26939985445622021077560944887, −1.56916728184634558329454248126, −0.69151166410011144602362313742, 0.6174047123595988832127805702, 1.91467533297884004855741592926, 2.90795007989563440389594085744, 3.991389016042531479187818614241, 5.06406102095239298763766697557, 5.740839910844680947942217394846, 6.41029148526828403364664039638, 7.371135216526130359383896368694, 8.45331072672882102278489291499, 9.41686572901223273780144014094, 10.24064918413611193350197641069, 10.946399539825165572250060228234, 11.561149790452246908769082766869, 12.30886668771347267526348549990, 13.3902302088706915060427273417, 14.30747938304513758774100715103, 15.12406155819234044029572391117, 15.486179857786118931368211346325, 16.83689828903813717402596358606, 17.29741054780107329004022705915, 17.88606497511294664689278035460, 18.8467252967545547831761831031, 19.29131668668482848401745053706, 20.866275639111500140525808406299, 21.23506522955105352667570155311

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