Properties

Label 1-847-847.472-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.498 - 0.866i$
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  + (−0.580 − 0.814i)2-s + (0.5 + 0.866i)3-s + (−0.327 + 0.945i)4-s + (−0.723 + 0.690i)5-s + (0.415 − 0.909i)6-s + (0.959 − 0.281i)8-s + (−0.5 + 0.866i)9-s + (0.981 + 0.189i)10-s + (−0.981 + 0.189i)12-s + (−0.654 − 0.755i)13-s + (−0.959 − 0.281i)15-s + (−0.786 − 0.618i)16-s + (0.0475 − 0.998i)17-s + (0.995 − 0.0950i)18-s + (0.0475 + 0.998i)19-s + (−0.415 − 0.909i)20-s + ⋯
L(s)  = 1  + (−0.580 − 0.814i)2-s + (0.5 + 0.866i)3-s + (−0.327 + 0.945i)4-s + (−0.723 + 0.690i)5-s + (0.415 − 0.909i)6-s + (0.959 − 0.281i)8-s + (−0.5 + 0.866i)9-s + (0.981 + 0.189i)10-s + (−0.981 + 0.189i)12-s + (−0.654 − 0.755i)13-s + (−0.959 − 0.281i)15-s + (−0.786 − 0.618i)16-s + (0.0475 − 0.998i)17-s + (0.995 − 0.0950i)18-s + (0.0475 + 0.998i)19-s + (−0.415 − 0.909i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.498 - 0.866i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (472, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.498 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1423088874 - 0.2461280564i\)
\(L(\frac12)\) \(\approx\) \(0.1423088874 - 0.2461280564i\)
\(L(1)\) \(\approx\) \(0.6190973443 + 0.007820059223i\)
\(L(1)\) \(\approx\) \(0.6190973443 + 0.007820059223i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.580 + 0.814i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.723 - 0.690i)T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (-0.0475 + 0.998i)T \)
19 \( 1 + (-0.0475 - 0.998i)T \)
23 \( 1 + (0.786 + 0.618i)T \)
29 \( 1 + (0.841 - 0.540i)T \)
31 \( 1 + (0.981 + 0.189i)T \)
37 \( 1 + (0.327 + 0.945i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.995 - 0.0950i)T \)
53 \( 1 + (0.786 - 0.618i)T \)
59 \( 1 + (0.580 - 0.814i)T \)
61 \( 1 + (0.995 + 0.0950i)T \)
67 \( 1 + (0.995 - 0.0950i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (-0.928 + 0.371i)T \)
79 \( 1 + (0.723 - 0.690i)T \)
83 \( 1 + (0.142 + 0.989i)T \)
89 \( 1 + (-0.888 + 0.458i)T \)
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

−22.70302126166446528731774921821, −21.57470948885183540952287545829, −20.307317703890732087352701971496, −19.74130440126291188407776603927, −19.18638086040065204498628820881, −18.47448691575682143093076273875, −17.41298300835867876578043554856, −16.97587521432863459602100459172, −15.93960737148892873577646028122, −15.17278717503271012297396209995, −14.47151781116524702993806725136, −13.54723593860128288303874354456, −12.78953410649988918392329803763, −11.85323415190331063385134589260, −10.98028577160596190484528522002, −9.51123775332140773608385605812, −9.06466815033561095393346593610, −8.07253581746257587388138123562, −7.59729907275856452819009922701, −6.73942127998450129184671931129, −5.8078737481795787764803262374, −4.70198374763653036807589375918, −3.71873297339472556454998960243, −2.13507145798639916116518390824, −1.217444756716266148849359890341, 0.1530715377931523152836108091, 2.0953119325195728731504825342, 2.927136774127422102530641521998, 3.68937908145295516682439031179, 4.44314181655353256187156633463, 5.64666018596849801574999347709, 7.46018787329360183444845322306, 7.6698746076710992048458089876, 8.835220435264242258890198404585, 9.55110254448962745433330609147, 10.55259035030197287487215055670, 10.809267360861434657631242811776, 11.95039315622753977962421482975, 12.600044652807815025255147301488, 13.96240328624408023634653845739, 14.49091701794966441917142062247, 15.547185601023462228149396729709, 16.224372143020166728701729384513, 17.01657210870434898685712609531, 18.15773440074145412681176666545, 18.74000269193280267189484465449, 19.63703576597552944619855608756, 20.231720758177567935538785979017, 20.78419187770056965962841279481, 21.81930582863431117646792388460

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