↑

Properties

Label 1-34e2-1156.543-r1-0-0
Degree $1$
Conductor $1156$
Sign $-0.978 + 0.205i$
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  + (−0.850 + 0.526i)3-s + (0.273 − 0.961i)5-s + (0.445 − 0.895i)7-s + (0.445 − 0.895i)9-s + (−0.982 + 0.183i)11-s + (−0.273 − 0.961i)13-s + (0.273 + 0.961i)15-s + (−0.0922 + 0.995i)19-s + (0.0922 + 0.995i)21-s + (0.445 − 0.895i)23-s + (−0.850 − 0.526i)25-s + (0.0922 + 0.995i)27-s + (0.982 − 0.183i)29-s + (−0.273 − 0.961i)31-s + (0.739 − 0.673i)33-s + ⋯
L(s)  = 1  + (−0.850 + 0.526i)3-s + (0.273 − 0.961i)5-s + (0.445 − 0.895i)7-s + (0.445 − 0.895i)9-s + (−0.982 + 0.183i)11-s + (−0.273 − 0.961i)13-s + (0.273 + 0.961i)15-s + (−0.0922 + 0.995i)19-s + (0.0922 + 0.995i)21-s + (0.445 − 0.895i)23-s + (−0.850 − 0.526i)25-s + (0.0922 + 0.995i)27-s + (0.982 − 0.183i)29-s + (−0.273 − 0.961i)31-s + (0.739 − 0.673i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.978 + 0.205i$
Analytic conductor: \(124.229\)
Root analytic conductor: \(124.229\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1156,\ (1:\ ),\ -0.978 + 0.205i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08494914431 - 0.8196635774i\)
\(L(\frac12)\) \(\approx\) \(-0.08494914431 - 0.8196635774i\)
\(L(1)\) \(\approx\) \(0.7341268736 - 0.2666511220i\)
\(L(1)\) \(\approx\) \(0.7341268736 - 0.2666511220i\)

Euler product

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

−21.54183102490151723947817547388, −21.22380596687254100885859861444, −19.57823751602861568842903818780, −19.04049453254819703509900086930, −18.25412701466790130420152277521, −17.86588306408884224385311642933, −17.10388292165742868785113224478, −15.97011882566904387109373070231, −15.44105642880338523813871278175, −14.42923113567623458299245521419, −13.66229823394418357979269171602, −12.87036583212370645331320032010, −11.898644506731770138563295629067, −11.3155411498635196742424882059, −10.68069175536656240743397763883, −9.75650809871924445162090894772, −8.70282552594509772662974038389, −7.67827455579881446868018775417, −6.914204531457345215877046736781, −6.22239504107457862337515381127, −5.29067964320294390457073301645, −4.69779232468834796090360475054, −3.04321322996366485099239492405, −2.31485885684207410785063883226, −1.37142585308456516041747023000, 0.2336458020284145063511814606, 0.850745397957291046393252461700, 2.11564209810104009519981369451, 3.59959862574835304859129643111, 4.476453903484930167860647880501, 5.17700901163242306787986149468, 5.75919075717858377495196606844, 6.94046093260927412265395178034, 7.917312377809832437873329573249, 8.64135572736555542088772419667, 10.01155146320752566400290646841, 10.19408625779807032381497482716, 11.08449339775837387542877405037, 12.12017180467343302980363716730, 12.76310588990405931780925988626, 13.44520822939675848865710164520, 14.57392364459808435524238252661, 15.37005642084487703849387676307, 16.34319617254092935233299335630, 16.65590703234949804359223672561, 17.70315866375371342627725008410, 17.860574019742418966384183763605, 19.14532561765309559080892749074, 20.34778603033093986004487002051, 20.70290349793148302613136660319

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