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Properties

Label 1-847-847.178-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.534 - 0.845i$
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  + (0.761 − 0.647i)2-s + (0.978 − 0.207i)3-s + (0.161 − 0.986i)4-s + (0.683 − 0.730i)5-s + (0.610 − 0.791i)6-s + (−0.516 − 0.856i)8-s + (0.913 − 0.406i)9-s + (0.0475 − 0.998i)10-s + (−0.0475 − 0.998i)12-s + (−0.998 − 0.0570i)13-s + (0.516 − 0.856i)15-s + (−0.948 − 0.318i)16-s + (−0.532 + 0.846i)17-s + (0.432 − 0.901i)18-s + (0.640 − 0.768i)19-s + (−0.610 − 0.791i)20-s + ⋯
L(s)  = 1  + (0.761 − 0.647i)2-s + (0.978 − 0.207i)3-s + (0.161 − 0.986i)4-s + (0.683 − 0.730i)5-s + (0.610 − 0.791i)6-s + (−0.516 − 0.856i)8-s + (0.913 − 0.406i)9-s + (0.0475 − 0.998i)10-s + (−0.0475 − 0.998i)12-s + (−0.998 − 0.0570i)13-s + (0.516 − 0.856i)15-s + (−0.948 − 0.318i)16-s + (−0.532 + 0.846i)17-s + (0.432 − 0.901i)18-s + (0.640 − 0.768i)19-s + (−0.610 − 0.791i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.534 - 0.845i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.534 - 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.646361016 - 2.989031724i\)
\(L(\frac12)\) \(\approx\) \(1.646361016 - 2.989031724i\)
\(L(1)\) \(\approx\) \(1.770768749 - 1.428795605i\)
\(L(1)\) \(\approx\) \(1.770768749 - 1.428795605i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.761 - 0.647i)T \)
3 \( 1 + (0.978 - 0.207i)T \)
5 \( 1 + (0.683 - 0.730i)T \)
13 \( 1 + (-0.998 - 0.0570i)T \)
17 \( 1 + (-0.532 + 0.846i)T \)
19 \( 1 + (0.640 - 0.768i)T \)
23 \( 1 + (0.580 + 0.814i)T \)
29 \( 1 + (-0.897 + 0.441i)T \)
31 \( 1 + (0.625 - 0.780i)T \)
37 \( 1 + (-0.710 - 0.703i)T \)
41 \( 1 + (-0.0285 + 0.999i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (0.432 + 0.901i)T \)
53 \( 1 + (-0.948 + 0.318i)T \)
59 \( 1 + (0.851 - 0.524i)T \)
61 \( 1 + (-0.179 + 0.983i)T \)
67 \( 1 + (0.723 + 0.690i)T \)
71 \( 1 + (0.696 + 0.717i)T \)
73 \( 1 + (-0.398 - 0.917i)T \)
79 \( 1 + (0.905 + 0.424i)T \)
83 \( 1 + (-0.870 - 0.491i)T \)
89 \( 1 + (0.786 + 0.618i)T \)
97 \( 1 + (-0.974 + 0.226i)T \)
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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

−22.39972798180551387713390580917, −21.69403211136121038034903811426, −20.85898479739506453843252109878, −20.354170394932505014419119282745, −19.1615386811878355462909277943, −18.38793814286103565656508228243, −17.473049577655660156725813402239, −16.62039376210584846664275208372, −15.66315532388982934424413204109, −14.99319106866352098436440855195, −14.21205231984389099425153462019, −13.870203067593481223043623657452, −12.95251260913227847134030255397, −12.07017018120764024706400263585, −10.90986824709862850910537080312, −9.88005961221341430231697253790, −9.15400956842590795489292151411, −8.11369629111830842059592512100, −7.20545866592208601522534457711, −6.69590215384618124407650946494, −5.42530536935988423623916088807, −4.650274852114888648559095261394, −3.507829339338678042948041142635, −2.748681204871757333475139008074, −2.00208778402817449060078479420, 1.07237342322098786597075023480, 2.027265230421178629625973356848, 2.74331987927712233885837644747, 3.859772508138294303387233151144, 4.74933116558834114183880232154, 5.588701588006716568681506268028, 6.70031761353960406484758189418, 7.66471340793703237469253709326, 8.96181467424912945773532178447, 9.44139578278822383821135549155, 10.23215352469993581106404831907, 11.35040708724148819669868073245, 12.44400145261594199611716910043, 12.96779835507090250602955248835, 13.61608625165729731680494740214, 14.364152845942549878897273145943, 15.16465743552476923782280343088, 15.875173593974247274121721303618, 17.17863530206858142060653418888, 17.94435023066044799653810167018, 19.102822674024400926584795645765, 19.60080703843926057042504832875, 20.38391374775617754559712517423, 20.91974698037205172266686781337, 21.796366926635338964511919128691

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