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Properties

Label 1-847-847.62-r0-0-0
Degree $1$
Conductor $847$
Sign $0.928 + 0.372i$
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  + (0.254 − 0.967i)2-s + (0.809 + 0.587i)3-s + (−0.870 − 0.491i)4-s + (0.985 + 0.170i)5-s + (0.774 − 0.633i)6-s + (−0.696 + 0.717i)8-s + (0.309 + 0.951i)9-s + (0.415 − 0.909i)10-s + (−0.415 − 0.909i)12-s + (−0.736 + 0.676i)13-s + (0.696 + 0.717i)15-s + (0.516 + 0.856i)16-s + (−0.0285 + 0.999i)17-s + (0.998 − 0.0570i)18-s + (0.610 + 0.791i)19-s + (−0.774 − 0.633i)20-s + ⋯
L(s)  = 1  + (0.254 − 0.967i)2-s + (0.809 + 0.587i)3-s + (−0.870 − 0.491i)4-s + (0.985 + 0.170i)5-s + (0.774 − 0.633i)6-s + (−0.696 + 0.717i)8-s + (0.309 + 0.951i)9-s + (0.415 − 0.909i)10-s + (−0.415 − 0.909i)12-s + (−0.736 + 0.676i)13-s + (0.696 + 0.717i)15-s + (0.516 + 0.856i)16-s + (−0.0285 + 0.999i)17-s + (0.998 − 0.0570i)18-s + (0.610 + 0.791i)19-s + (−0.774 − 0.633i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.928 + 0.372i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.928 + 0.372i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.107732024 + 0.4071780808i\)
\(L(\frac12)\) \(\approx\) \(2.107732024 + 0.4071780808i\)
\(L(1)\) \(\approx\) \(1.550439190 - 0.1302422410i\)
\(L(1)\) \(\approx\) \(1.550439190 - 0.1302422410i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.254 - 0.967i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.985 + 0.170i)T \)
13 \( 1 + (-0.736 + 0.676i)T \)
17 \( 1 + (-0.0285 + 0.999i)T \)
19 \( 1 + (0.610 + 0.791i)T \)
23 \( 1 + (-0.654 + 0.755i)T \)
29 \( 1 + (-0.941 + 0.336i)T \)
31 \( 1 + (-0.993 - 0.113i)T \)
37 \( 1 + (0.198 + 0.980i)T \)
41 \( 1 + (-0.362 - 0.931i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (0.998 + 0.0570i)T \)
53 \( 1 + (0.516 - 0.856i)T \)
59 \( 1 + (0.362 - 0.931i)T \)
61 \( 1 + (-0.254 - 0.967i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (-0.564 + 0.825i)T \)
73 \( 1 + (0.0855 - 0.996i)T \)
79 \( 1 + (-0.897 + 0.441i)T \)
83 \( 1 + (-0.921 + 0.389i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (0.985 - 0.170i)T \)
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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

−22.15389213648594823770310590383, −21.38206606481420857287687110645, −20.4385125282976054675125004529, −19.78493084997824437185363159578, −18.35437773026879796658687799948, −18.20332718325086383817167241603, −17.304249871467887690325490288377, −16.44491593919701429167714663256, −15.49992579828748673785958323529, −14.587342337285065191856751711814, −14.129227600842350110507660658457, −13.23202003585113762860390807070, −12.823788491920411936678576846290, −11.81251588014442580280551369801, −10.17865728170577435835191193189, −9.33166428858643053367198374668, −8.871140960290838257503836246663, −7.6572382566737987119750243817, −7.204984877544587233241754381273, −6.154000790096128373047852865489, −5.37369831038000183142806277603, −4.375632625189131758857982426934, −3.069844110924640008684424980931, −2.31169640750009246691129178164, −0.78445571172673937876734419788, 1.74252456389807675303288472253, 2.08859247323549563894299321531, 3.33346748978356449832517521144, 3.98803971109269370315399239937, 5.14338198490186626825911835541, 5.80967740495857918914305597468, 7.2566343544512349482940335116, 8.4506716433651272931356943996, 9.2773552534935722985371553053, 9.89182111116389100723291557594, 10.46077090123941987031189812052, 11.4481946874417035215889165891, 12.540527238745277515319711685890, 13.310013814162247980079009475, 14.14004701483336539659554983491, 14.50945003669779575731781932849, 15.43536377802675816432911413888, 16.720146092728053300084236008772, 17.3970754629108820956714253513, 18.57036308842574932923049285284, 19.00583861059717883301381431676, 20.06784748767918391040662264987, 20.540707135773765639171017103926, 21.40563553521568525345679953062, 21.99011568651644207589936720026

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