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Properties

Label 1-891-891.203-r1-0-0
Degree $1$
Conductor $891$
Sign $-0.974 + 0.222i$
Analytic cond. $95.7512$
Root an. cond. $95.7512$
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  + (−0.735 + 0.677i)2-s + (0.0813 − 0.996i)4-s + (0.459 + 0.888i)5-s + (−0.999 − 0.0232i)7-s + (0.615 + 0.788i)8-s + (−0.939 − 0.342i)10-s + (−0.0116 + 0.999i)13-s + (0.750 − 0.660i)14-s + (−0.986 − 0.162i)16-s + (0.997 − 0.0697i)17-s + (−0.374 + 0.927i)19-s + (0.922 − 0.385i)20-s + (0.286 − 0.957i)23-s + (−0.578 + 0.815i)25-s + (−0.669 − 0.743i)26-s + ⋯
L(s)  = 1  + (−0.735 + 0.677i)2-s + (0.0813 − 0.996i)4-s + (0.459 + 0.888i)5-s + (−0.999 − 0.0232i)7-s + (0.615 + 0.788i)8-s + (−0.939 − 0.342i)10-s + (−0.0116 + 0.999i)13-s + (0.750 − 0.660i)14-s + (−0.986 − 0.162i)16-s + (0.997 − 0.0697i)17-s + (−0.374 + 0.927i)19-s + (0.922 − 0.385i)20-s + (0.286 − 0.957i)23-s + (−0.578 + 0.815i)25-s + (−0.669 − 0.743i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.974 + 0.222i$
Analytic conductor: \(95.7512\)
Root analytic conductor: \(95.7512\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 891,\ (1:\ ),\ -0.974 + 0.222i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1286093667 + 1.142612584i\)
\(L(\frac12)\) \(\approx\) \(0.1286093667 + 1.142612584i\)
\(L(1)\) \(\approx\) \(0.6354499150 + 0.4229215678i\)
\(L(1)\) \(\approx\) \(0.6354499150 + 0.4229215678i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.735 + 0.677i)T \)
5 \( 1 + (0.459 + 0.888i)T \)
7 \( 1 + (-0.999 - 0.0232i)T \)
13 \( 1 + (-0.0116 + 0.999i)T \)
17 \( 1 + (0.997 - 0.0697i)T \)
19 \( 1 + (-0.374 + 0.927i)T \)
23 \( 1 + (0.286 - 0.957i)T \)
29 \( 1 + (0.750 + 0.660i)T \)
31 \( 1 + (0.982 + 0.185i)T \)
37 \( 1 + (0.990 + 0.139i)T \)
41 \( 1 + (0.196 + 0.980i)T \)
43 \( 1 + (-0.835 - 0.549i)T \)
47 \( 1 + (0.903 + 0.427i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (0.967 + 0.253i)T \)
61 \( 1 + (-0.651 - 0.758i)T \)
67 \( 1 + (0.396 + 0.918i)T \)
71 \( 1 + (-0.559 - 0.829i)T \)
73 \( 1 + (0.0348 + 0.999i)T \)
79 \( 1 + (0.735 - 0.677i)T \)
83 \( 1 + (-0.871 - 0.489i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.459 + 0.888i)T \)
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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

−21.26786030750009694133509481012, −20.53512722504030147628383840547, −19.696112723521539254021935030128, −19.28746936965755434053879534569, −18.228573535068739284820745033941, −17.362835957815197918463939493619, −16.919654796126854232345579415888, −15.97812243263626434207550035634, −15.35688225137427141637542034680, −13.722077529771543383972087928540, −13.14197087544549211850456064596, −12.47991844945260559272651966472, −11.7556880932464357712747662850, −10.56398391929650599793469321237, −9.85138739147148607642805681067, −9.295248114142228235096704871382, −8.37977597642619059926231798717, −7.59245227257810049929552751654, −6.43617840527689013444043901321, −5.46258591782112610343639676490, −4.31275441712381919167359543142, −3.22093340698625191670490223751, −2.44146519975332131962056465265, −1.09178688081507638629024458553, −0.4040158068046145147559449362, 1.065948110820410250388859150048, 2.27176783486656602203880074459, 3.26427597031342607373999707455, 4.58986531552399690237975871073, 5.88263454855956983324695210939, 6.44066833095039111821100643189, 7.062739009369078894761151844182, 8.09454679676970619041026064136, 9.086663830185901174743518530872, 9.96980219043715779234177962963, 10.3001992315055434120828635859, 11.37986393708615939979145320827, 12.44261080489705655889705652402, 13.60713289182746857256121336208, 14.34243441733852176180053497850, 14.88085880181469550894654345930, 16.001154889505115897936656401821, 16.56416281361594823963699954521, 17.26702895283293735044693983510, 18.34976605063033055799422326579, 18.850381952463295653873347996649, 19.32700817856947894518506173658, 20.42129234156550283908266380868, 21.41274669319934715734636086841, 22.27611648218806035253022515513

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