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Properties

Label 1-42e2-1764.1051-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: $\chi_{1764} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ 0.959 + 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.880180530 + 0.4131555802i\)
\(L(\frac12)\) \(\approx\) \(2.880180530 + 0.4131555802i\)
\(L(1)\) \(\approx\) \(1.353846732 + 0.0004472472439i\)
\(L(1)\) \(\approx\) \(1.353846732 + 0.0004472472439i\)

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

−19.97467609727463526372575363217, −19.34193085973756211847763236125, −18.412306948746259157863142170983, −17.74903833458316491088466133742, −17.33531758590584272448477517201, −16.304539267278348792869781148813, −15.616162193012317352060070080209, −14.65490185329807596298834535561, −14.120550741845060375809458065553, −13.45820864487635389948334266491, −12.58743992061517790386539946019, −11.76716816463567098001157849944, −10.858862925837394195090353163761, −10.291266350846420000527952889662, −9.438250033334994985170488665153, −8.745031780646679976073594477428, −7.79884441460019531560554843668, −6.769359023781702173468382516170, −6.25534495319847363358982702034, −5.44088598380350596441908776378, −4.4366015514014805169763511007, −3.41300784752420934320571485448, −2.62420615769942527505160806158, −1.66920768813599468074293933718, −0.63784737153927320069781458752, 0.82409805238920913521353200516, 1.77625609239731104492574960343, 2.38235685059047805668835308601, 3.98499894011337997447374224959, 4.28969926147145009498567025034, 5.46887182067813503439699456971, 6.4400033880705313865309787073, 6.620736577787529672729917212942, 8.21798191897272118721704733270, 8.63761867033096334966602514313, 9.52894153369150937013908378498, 10.14355249894757414798471993037, 11.064967776826290639776986221387, 12.00885484766449625560971344919, 12.55854661613901122753525660423, 13.520717796800285070513762300367, 14.03216545051149079759256543347, 14.81271857118928071556120427971, 15.70063465122478972241707346357, 16.636249509976627809032854540052, 17.12087984883442674891049956859, 17.67370943854819965707631410623, 18.6915682950318959887403207212, 19.36202317248407068131531970012, 20.11750827525428836889512410448

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