Properties

Label 2-42e2-28.27-c1-0-30
Degree $2$
Conductor $1764$
Sign $0.983 - 0.178i$
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  + (0.988 − 1.01i)2-s + (−0.0442 − 1.99i)4-s + 1.12i·5-s + (−2.06 − 1.93i)8-s + (1.14 + 1.11i)10-s + 1.35i·11-s + 5.58i·13-s + (−3.99 + 0.177i)16-s + 3.97i·17-s + 6.14·19-s + (2.25 − 0.0500i)20-s + (1.36 + 1.33i)22-s + 6.61i·23-s + 3.72·25-s + (5.64 + 5.52i)26-s + ⋯
L(s)  = 1  + (0.699 − 0.714i)2-s + (−0.0221 − 0.999i)4-s + 0.504i·5-s + (−0.730 − 0.683i)8-s + (0.360 + 0.353i)10-s + 0.407i·11-s + 1.54i·13-s + (−0.999 + 0.0442i)16-s + 0.963i·17-s + 1.40·19-s + (0.504 − 0.0111i)20-s + (0.290 + 0.284i)22-s + 1.37i·23-s + 0.745·25-s + (1.10 + 1.08i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.983 - 0.178i$
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: \((2,\ 1764,\ (\ :1/2),\ 0.983 - 0.178i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.262154798\)
\(L(\frac12)\) \(\approx\) \(2.262154798\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.988 + 1.01i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 1.12iT - 5T^{2} \)
11 \( 1 - 1.35iT - 11T^{2} \)
13 \( 1 - 5.58iT - 13T^{2} \)
17 \( 1 - 3.97iT - 17T^{2} \)
19 \( 1 - 6.14T + 19T^{2} \)
23 \( 1 - 6.61iT - 23T^{2} \)
29 \( 1 + 8.55T + 29T^{2} \)
31 \( 1 + 2.92T + 31T^{2} \)
37 \( 1 - 6.32T + 37T^{2} \)
41 \( 1 + 0.149iT - 41T^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 - 3.59T + 47T^{2} \)
53 \( 1 + 0.733T + 53T^{2} \)
59 \( 1 - 14.4T + 59T^{2} \)
61 \( 1 - 9.11iT - 61T^{2} \)
67 \( 1 - 0.234iT - 67T^{2} \)
71 \( 1 + 8.74iT - 71T^{2} \)
73 \( 1 - 1.76iT - 73T^{2} \)
79 \( 1 - 8.40iT - 79T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 + 7.92iT - 89T^{2} \)
97 \( 1 - 6.23iT - 97T^{2} \)
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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

−9.472208057287002699100102514282, −8.883186047671831346699212895277, −7.41740976378390683169847439056, −6.94192093622769688717309109901, −5.88901713474394511821802716697, −5.22600884758472897730616225913, −4.07292054362374173647955351397, −3.54596506373356432775792288875, −2.31341819715209911230591892701, −1.43517920561097840851495720472, 0.70123594313566234270088409998, 2.67521262362641180170451339780, 3.37622099366267173595870610116, 4.52666990481256582459033631194, 5.32550399670735450351902808922, 5.80238557959193709756873015022, 6.91048841715987466961583600637, 7.66529752161647686645730123201, 8.265009574830218944974394887135, 9.105272105110329495048764746042

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