Properties

Label 1-42e2-1764.1091-r1-0-0
Degree $1$
Conductor $1764$
Sign $-0.693 + 0.720i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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  + (−0.733 − 0.680i)5-s + (0.826 + 0.563i)11-s + (−0.0747 + 0.997i)13-s + (0.623 + 0.781i)17-s + 19-s + (0.365 + 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.365 + 0.930i)29-s + (−0.5 + 0.866i)31-s + (0.623 + 0.781i)37-s + (−0.733 − 0.680i)41-s + (0.733 − 0.680i)43-s + (−0.826 − 0.563i)47-s + (−0.623 + 0.781i)53-s + (−0.222 − 0.974i)55-s + ⋯
L(s)  = 1  + (−0.733 − 0.680i)5-s + (0.826 + 0.563i)11-s + (−0.0747 + 0.997i)13-s + (0.623 + 0.781i)17-s + 19-s + (0.365 + 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.365 + 0.930i)29-s + (−0.5 + 0.866i)31-s + (0.623 + 0.781i)37-s + (−0.733 − 0.680i)41-s + (0.733 − 0.680i)43-s + (−0.826 − 0.563i)47-s + (−0.623 + 0.781i)53-s + (−0.222 − 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.693 + 0.720i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ -0.693 + 0.720i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4898280688 + 1.150542832i\)
\(L(\frac12)\) \(\approx\) \(0.4898280688 + 1.150542832i\)
\(L(1)\) \(\approx\) \(0.9484969165 + 0.1494912109i\)
\(L(1)\) \(\approx\) \(0.9484969165 + 0.1494912109i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.733 - 0.680i)T \)
11 \( 1 + (0.826 + 0.563i)T \)
13 \( 1 + (-0.0747 + 0.997i)T \)
17 \( 1 + (0.623 + 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (0.365 + 0.930i)T \)
29 \( 1 + (-0.365 + 0.930i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (-0.733 - 0.680i)T \)
43 \( 1 + (0.733 - 0.680i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (-0.623 + 0.781i)T \)
59 \( 1 + (-0.955 - 0.294i)T \)
61 \( 1 + (-0.365 + 0.930i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.0747 - 0.997i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−19.83671137367128331863565458803, −18.95350903653621770665095026134, −18.45625576606100980236467110110, −17.68965368999388226327116900460, −16.69679464072916590717134296755, −16.11859845617338203575433900044, −15.291622621529222942751053215801, −14.57979453472515918664305290029, −14.03891886782024198980612791729, −13.02805572176373599013276223236, −12.201490847053138963849216034580, −11.39628627892852508705152822295, −10.99630255371177494272777481686, −9.89868729259018238858913987194, −9.29663196318355135164530486423, −8.05545238551793558197532540431, −7.718155741623564893069960278043, −6.72852938237861308856153446385, −5.97176174356266391450561549963, −5.018847198292431589489792227172, −3.97997552970734074545766557275, −3.23477388793726904538408610450, −2.5614692218011333503046352132, −1.04623189282711208191740892043, −0.264085272841736650086086855176, 1.20688242105391870910409786700, 1.6940247439468892009611860269, 3.29671747077481066376442794234, 3.867114332725816462580953173629, 4.788154293470230752008086321862, 5.487269341917555801669140102249, 6.6786309149457678342360010229, 7.33646998033621510395235679324, 8.12764725890647820371083455693, 9.13861056235894444072516026876, 9.44355453208339751828594072226, 10.631987052217126958162940781099, 11.53742880319330144344473131373, 12.09319797224078432031164855708, 12.6667368972642993331191080027, 13.69755489148827550074825330327, 14.420781626470284008923515209976, 15.210280143430936402860762533379, 15.91046573905222027287804356629, 16.81578941562106762530621229689, 17.062049540459150215417644618033, 18.21429215136588822635156886929, 18.95400333296154958683950103022, 19.77100892392689809694287035996, 20.09500546244262733115867646656

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