Properties

Label 1-847-847.370-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.427 + 0.904i$
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  + (0.921 + 0.389i)2-s + (0.809 + 0.587i)3-s + (0.696 + 0.717i)4-s + (0.254 + 0.967i)5-s + (0.516 + 0.856i)6-s + (0.362 + 0.931i)8-s + (0.309 + 0.951i)9-s + (−0.142 + 0.989i)10-s + (0.142 + 0.989i)12-s + (0.897 − 0.441i)13-s + (−0.362 + 0.931i)15-s + (−0.0285 + 0.999i)16-s + (−0.736 − 0.676i)17-s + (−0.0855 + 0.996i)18-s + (0.198 − 0.980i)19-s + (−0.516 + 0.856i)20-s + ⋯
L(s)  = 1  + (0.921 + 0.389i)2-s + (0.809 + 0.587i)3-s + (0.696 + 0.717i)4-s + (0.254 + 0.967i)5-s + (0.516 + 0.856i)6-s + (0.362 + 0.931i)8-s + (0.309 + 0.951i)9-s + (−0.142 + 0.989i)10-s + (0.142 + 0.989i)12-s + (0.897 − 0.441i)13-s + (−0.362 + 0.931i)15-s + (−0.0285 + 0.999i)16-s + (−0.736 − 0.676i)17-s + (−0.0855 + 0.996i)18-s + (0.198 − 0.980i)19-s + (−0.516 + 0.856i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.427 + 0.904i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (370, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.427 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.981751597 + 3.128123779i\)
\(L(\frac12)\) \(\approx\) \(1.981751597 + 3.128123779i\)
\(L(1)\) \(\approx\) \(1.941753298 + 1.465420210i\)
\(L(1)\) \(\approx\) \(1.941753298 + 1.465420210i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.921 + 0.389i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.254 + 0.967i)T \)
13 \( 1 + (0.897 - 0.441i)T \)
17 \( 1 + (-0.736 - 0.676i)T \)
19 \( 1 + (0.198 - 0.980i)T \)
23 \( 1 + (-0.959 + 0.281i)T \)
29 \( 1 + (0.870 + 0.491i)T \)
31 \( 1 + (0.985 - 0.170i)T \)
37 \( 1 + (-0.466 - 0.884i)T \)
41 \( 1 + (0.974 - 0.226i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (-0.0855 - 0.996i)T \)
53 \( 1 + (-0.0285 - 0.999i)T \)
59 \( 1 + (-0.974 - 0.226i)T \)
61 \( 1 + (-0.921 + 0.389i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (0.993 - 0.113i)T \)
73 \( 1 + (0.610 - 0.791i)T \)
79 \( 1 + (-0.774 - 0.633i)T \)
83 \( 1 + (-0.564 + 0.825i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 + (0.254 - 0.967i)T \)
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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

−21.44874497751820408457879558417, −21.14836338480787945233471207754, −20.246440960547670469725023177238, −19.8052431423832949193098396553, −18.90691283746433335640940988386, −18.09227990030425997588880801418, −16.96893691323240467042819545295, −15.90556727399976895235877422858, −15.38117266646372728978551015480, −14.096344808733327401201882794111, −13.84571537128535306537606796987, −12.95108103875063742302143304195, −12.338361113884779063903188406186, −11.61741320642034352358894857258, −10.35516806220843683027715388955, −9.51649910121290163263819418468, −8.52374460192545890387143396019, −7.833649364527542035961437876611, −6.35682278791820204650077751715, −6.08969036027009386567663749992, −4.59270029708112059715620803696, −4.00399326542895620086799590062, −2.90305973224059892772275097542, −1.799487754761810844068042778179, −1.20877479278823641173155333659, 2.01635916259892473890324939678, 2.83354753437556992390962877989, 3.53239777048413977237792924583, 4.45728235360347304596213951879, 5.43873603063127561856913600627, 6.47085183182182381317029529891, 7.24513029635384850560959851551, 8.16023203767221808392401606378, 9.05464626049550005743438319229, 10.22142330315388957876852764466, 10.931066454469355274792818521, 11.72364707676956480393053121370, 13.04347269562719757822403194415, 13.83585153095250006990173920057, 14.074648398949299416519596803008, 15.273871712925262520050732815240, 15.55324757941978822678727565944, 16.32939473694516307254786016100, 17.612616136750705857576928063996, 18.23630784300682315864147205334, 19.53806907732994009257336887958, 20.078666387875990356055009659, 21.050143445373818743018187849627, 21.59290033490362708929486692339, 22.37553616103508501183544922693

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