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Properties

Label 2-42e2-28.27-c1-0-75
Degree $2$
Conductor $1764$
Sign $0.912 + 0.409i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 2.00·4-s + 0.317i·5-s + 2.82·8-s + 0.448i·10-s − 4.01i·13-s + 4.00·16-s − 6.62i·17-s + 0.634i·20-s + 4.89·25-s − 5.67i·26-s + 4.24·29-s + 5.65·32-s − 9.37i·34-s + 9.89·37-s + ⋯
L(s)  = 1  + 1.00·2-s + 1.00·4-s + 0.141i·5-s + 1.00·8-s + 0.141i·10-s − 1.11i·13-s + 1.00·16-s − 1.60i·17-s + 0.141i·20-s + 0.979·25-s − 1.11i·26-s + 0.787·29-s + 1.00·32-s − 1.60i·34-s + 1.62·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.912 + 0.409i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.912 + 0.409i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.534411239\)
\(L(\frac12)\) \(\approx\) \(3.534411239\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 0.317iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 4.01iT - 13T^{2} \)
17 \( 1 + 6.62iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 9.89T + 37T^{2} \)
41 \( 1 - 12.3iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 14T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13.8iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 12.4iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 18.6iT - 89T^{2} \)
97 \( 1 + 14.2iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.397396106094102875033163944754, −8.178213162936576937712577902345, −7.55482942919667576061411815349, −6.70602851958225345087898508586, −5.95995759557320670965471995088, −5.02406697975640140912464137891, −4.46222885171264332616050298650, −3.07712191370190852387966848084, −2.72975526588954003770628498149, −1.06475874124527926433392727957, 1.44741010904567748042583323160, 2.46397283596897964060522477919, 3.64684926089731551059715536575, 4.34895787961738485338694836577, 5.16572118657436220999880327749, 6.21856334523532731121864236337, 6.63611348926093800298847039189, 7.66149374840494825318246299719, 8.447668942514759685934328351035, 9.359968277386162852109057991597

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