Properties

Label 1-847-847.811-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.570 + 0.821i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
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  + (0.736 + 0.676i)2-s + (0.809 − 0.587i)3-s + (0.0855 + 0.996i)4-s + (0.0285 + 0.999i)5-s + (0.993 + 0.113i)6-s + (−0.610 + 0.791i)8-s + (0.309 − 0.951i)9-s + (−0.654 + 0.755i)10-s + (0.654 + 0.755i)12-s + (−0.921 + 0.389i)13-s + (0.610 + 0.791i)15-s + (−0.985 + 0.170i)16-s + (−0.254 + 0.967i)17-s + (0.870 − 0.491i)18-s + (−0.362 + 0.931i)19-s + (−0.993 + 0.113i)20-s + ⋯
L(s)  = 1  + (0.736 + 0.676i)2-s + (0.809 − 0.587i)3-s + (0.0855 + 0.996i)4-s + (0.0285 + 0.999i)5-s + (0.993 + 0.113i)6-s + (−0.610 + 0.791i)8-s + (0.309 − 0.951i)9-s + (−0.654 + 0.755i)10-s + (0.654 + 0.755i)12-s + (−0.921 + 0.389i)13-s + (0.610 + 0.791i)15-s + (−0.985 + 0.170i)16-s + (−0.254 + 0.967i)17-s + (0.870 − 0.491i)18-s + (−0.362 + 0.931i)19-s + (−0.993 + 0.113i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.570 + 0.821i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.570 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.145618835 + 2.191828715i\)
\(L(\frac12)\) \(\approx\) \(1.145618835 + 2.191828715i\)
\(L(1)\) \(\approx\) \(1.509822994 + 0.9833237734i\)
\(L(1)\) \(\approx\) \(1.509822994 + 0.9833237734i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.736 - 0.676i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.0285 - 0.999i)T \)
13 \( 1 + (0.921 - 0.389i)T \)
17 \( 1 + (0.254 - 0.967i)T \)
19 \( 1 + (0.362 - 0.931i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (-0.998 - 0.0570i)T \)
31 \( 1 + (0.516 + 0.856i)T \)
37 \( 1 + (-0.974 + 0.226i)T \)
41 \( 1 + (-0.198 - 0.980i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.870 - 0.491i)T \)
53 \( 1 + (0.985 + 0.170i)T \)
59 \( 1 + (0.198 - 0.980i)T \)
61 \( 1 + (0.736 - 0.676i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
71 \( 1 + (-0.774 - 0.633i)T \)
73 \( 1 + (-0.696 + 0.717i)T \)
79 \( 1 + (-0.564 + 0.825i)T \)
83 \( 1 + (-0.897 + 0.441i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (-0.0285 + 0.999i)T \)
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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

−21.76176953772312615897308185564, −20.958560109287667348998039426826, −20.25516654185316348153657828016, −19.85534096932247267920818594365, −19.09686300996296063069305588916, −17.962678553458227013266337534988, −16.84794032381007792712349106026, −15.85884263974902564406731592410, −15.40731682323949701563260738920, −14.31051088490522216982416619466, −13.83672181205709921951440055122, −12.81477566561188707509063317853, −12.35533022610107728363142179110, −11.19044944488901542643430480062, −10.34228203571233066781843508150, −9.4504924036737629509652037342, −8.94701902684050481812857749060, −7.84688021044285697853909932036, −6.66408517017119877333161763144, −5.22701939014351834781399907432, −4.79145936878621185106118552985, −4.01594010339484556467447252729, −2.79084831532007568598048237272, −2.20690882636900363031257489589, −0.7385421281789049090982459569, 1.90103176391968029513055563035, 2.684483743129842152871019349474, 3.618482060154548603451816602447, 4.38820035181730403251191841338, 5.95106505913423491707596073569, 6.42260358382092027441327482970, 7.55032824150292709907595732630, 7.77647384679496171446058513045, 8.9883691927550168450054458949, 9.93817519892806808861342999846, 11.16006531494953343357687977116, 12.13185331117631337533015402268, 12.802171348273802202027042121661, 13.750583707607931309589829397622, 14.35798241942731687496631617938, 14.94484692144369716990656198346, 15.55630439198298621278142938275, 16.786077176510911934171773993607, 17.60127926737656903483079547133, 18.33979762523014486172558542765, 19.270440559314947730161835558341, 19.885148843108898124960359055575, 21.07557524313602756304827860916, 21.6454055311322070598833595508, 22.41433612742652311509696394268

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