Properties

Label 1-42e2-1764.167-r1-0-0
Degree $1$
Conductor $1764$
Sign $0.959 - 0.281i$
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  + (0.826 + 0.563i)5-s + (−0.988 + 0.149i)11-s + (−0.365 + 0.930i)13-s + (−0.222 − 0.974i)17-s + 19-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.955 − 0.294i)29-s + (−0.5 − 0.866i)31-s + (−0.222 − 0.974i)37-s + (0.826 + 0.563i)41-s + (−0.826 + 0.563i)43-s + (0.988 − 0.149i)47-s + (0.222 − 0.974i)53-s + (−0.900 − 0.433i)55-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)5-s + (−0.988 + 0.149i)11-s + (−0.365 + 0.930i)13-s + (−0.222 − 0.974i)17-s + 19-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.955 − 0.294i)29-s + (−0.5 − 0.866i)31-s + (−0.222 − 0.974i)37-s + (0.826 + 0.563i)41-s + (−0.826 + 0.563i)43-s + (0.988 − 0.149i)47-s + (0.222 − 0.974i)53-s + (−0.900 − 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.959 - 0.281i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ 0.959 - 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.183947701 - 0.3132825078i\)
\(L(\frac12)\) \(\approx\) \(2.183947701 - 0.3132825078i\)
\(L(1)\) \(\approx\) \(1.167939573 + 0.06799350175i\)
\(L(1)\) \(\approx\) \(1.167939573 + 0.06799350175i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.826 + 0.563i)T \)
11 \( 1 + (-0.988 + 0.149i)T \)
13 \( 1 + (-0.365 + 0.930i)T \)
17 \( 1 + (-0.222 - 0.974i)T \)
19 \( 1 + T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (-0.955 - 0.294i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (0.826 + 0.563i)T \)
43 \( 1 + (-0.826 + 0.563i)T \)
47 \( 1 + (0.988 - 0.149i)T \)
53 \( 1 + (0.222 - 0.974i)T \)
59 \( 1 + (-0.0747 - 0.997i)T \)
61 \( 1 + (-0.955 - 0.294i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.365 - 0.930i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
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

−20.24774862286814668341291678054, −19.44137179730595310687868940499, −18.40856375881080151338664099868, −17.93274421379365588720439343188, −17.1032423661118818434055862751, −16.581783180755679837430907902517, −15.54715664341550045515818329931, −15.06712154240143165572031253799, −13.96785618259908208740761602163, −13.332982009690591526154061795778, −12.739797107830529511618265822590, −12.06744792445244094756564748658, −10.73130527576551550065205802585, −10.45974506661282554967198739322, −9.43378320255434811995056614342, −8.79655385066352358842096931810, −7.88705397292124224972190420776, −7.16847931835308562818897863744, −5.97374335073939242241347553759, −5.389972100868626887426042684, −4.80152059407953380620578792569, −3.48165099062316057674447782158, −2.681650847864619739101842545039, −1.6716041235520171366296266079, −0.7505650079597498768970507991, 0.49867820279867258034115441737, 1.85129991844534706078952276558, 2.524599986741204073428268137318, 3.34590381948590878070532940596, 4.60633973891529645465590482493, 5.333424746734582374187160480088, 6.09467357431069712637815409680, 7.22173742963452491022566916803, 7.425825886466515550485279316345, 8.83771572714340381941941771984, 9.535903095660855105364687544986, 10.06582600170958527093041708792, 11.13647446926574234989552517266, 11.51872422529408033674512337100, 12.79474403926493890352430137698, 13.30505224087686219991392860409, 14.1380766461338125928786010923, 14.680657962203286398203232892420, 15.60600183862200655774974039586, 16.37091930326336885022660813534, 17.11214348088241856000854130584, 17.95416101012624306443564640624, 18.5240382569544165157077870061, 19.02899520223560143387511950828, 20.204852460089256166044703496872

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