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Properties

Label 1-891-891.812-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: $\chi_{891} (812, \cdot )$
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

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

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