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Properties

Label 1-847-847.615-r0-0-0
Degree $1$
Conductor $847$
Sign $0.604 - 0.796i$
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  + (−0.415 − 0.909i)2-s − 3-s + (−0.654 + 0.755i)4-s + (0.959 − 0.281i)5-s + (0.415 + 0.909i)6-s + (0.959 + 0.281i)8-s + 9-s + (−0.654 − 0.755i)10-s + (0.654 − 0.755i)12-s + (−0.654 + 0.755i)13-s + (−0.959 + 0.281i)15-s + (−0.142 − 0.989i)16-s + (0.841 − 0.540i)17-s + (−0.415 − 0.909i)18-s + (0.841 + 0.540i)19-s + (−0.415 + 0.909i)20-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)2-s − 3-s + (−0.654 + 0.755i)4-s + (0.959 − 0.281i)5-s + (0.415 + 0.909i)6-s + (0.959 + 0.281i)8-s + 9-s + (−0.654 − 0.755i)10-s + (0.654 − 0.755i)12-s + (−0.654 + 0.755i)13-s + (−0.959 + 0.281i)15-s + (−0.142 − 0.989i)16-s + (0.841 − 0.540i)17-s + (−0.415 − 0.909i)18-s + (0.841 + 0.540i)19-s + (−0.415 + 0.909i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.604 - 0.796i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (615, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.604 - 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8549213067 - 0.4244788478i\)
\(L(\frac12)\) \(\approx\) \(0.8549213067 - 0.4244788478i\)
\(L(1)\) \(\approx\) \(0.7118027606 - 0.2826493538i\)
\(L(1)\) \(\approx\) \(0.7118027606 - 0.2826493538i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.415 - 0.909i)T \)
3 \( 1 - T \)
5 \( 1 + (0.959 - 0.281i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (-0.142 - 0.989i)T \)
29 \( 1 + (-0.841 - 0.540i)T \)
31 \( 1 + (0.654 + 0.755i)T \)
37 \( 1 + (-0.654 - 0.755i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (-0.415 + 0.909i)T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (0.959 - 0.281i)T \)
83 \( 1 + (-0.142 + 0.989i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.959 + 0.281i)T \)
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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

−22.41347053397699331226243615906, −21.774223792121959946367516127117, −20.77844274158830011694183015381, −19.53685801433981870593922157901, −18.65469879978415178658503890460, −18.00728894776775526116383741200, −17.228545105870988348621391931572, −17.00299148041636521848451603221, −15.87093177552634123182953204161, −15.194217549168514946069975090231, −14.26890092869355620591547605971, −13.39168806680997534315931208281, −12.65165311451646578980316858944, −11.4702672923846907785603870800, −10.44349010206558347346466196908, −9.92999758285841976345587709596, −9.21076201248569866197873985674, −7.819701945091256410400238461707, −7.14619111752971549539892984301, −6.24402697057352732312286901409, −5.427767334652736837772219753, −5.087198642123074995311064851080, −3.58233323491656011894485912883, −1.927001856026559568837425765036, −0.833818089151263167006978828230, 0.88200906827333627653592169864, 1.77021320977441077932028910788, 2.82766381352585314871912157732, 4.21762206664225023499628307504, 5.00268495448212881736154244846, 5.85290530058226581563902919913, 6.97629663385027759914189763164, 7.91756640071497206907989690570, 9.2722748783349040574247867975, 9.72508040120611691063016858887, 10.47997025040049650775094765221, 11.36210795006969805331956483979, 12.25884040284379736735588620350, 12.62960198150401938231409190886, 13.77814345025073843854251456958, 14.3784363052246341942006821450, 16.08546219316828608512066435343, 16.65128192875111391247785586273, 17.28737343528554738369841572738, 18.07525482196784493683172838982, 18.63284586822331412011220620692, 19.49355285183449565772798744999, 20.793790853894207232292368034118, 21.016546783072481665786243520195, 21.96715854738155889480500196612

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