Properties

Label 2-42e2-63.20-c1-0-22
Degree $2$
Conductor $1764$
Sign $-0.329 - 0.944i$
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  + (1.09 + 1.34i)3-s + (1.95 + 3.39i)5-s + (−0.606 + 2.93i)9-s + (3.19 + 1.84i)11-s + (−0.480 + 0.277i)13-s + (−2.41 + 6.33i)15-s + 5.83·17-s − 5.33i·19-s + (1.96 − 1.13i)23-s + (−5.16 + 8.94i)25-s + (−4.60 + 2.40i)27-s + (3.53 + 2.04i)29-s + (7.00 − 4.04i)31-s + (1.01 + 6.31i)33-s − 7.79·37-s + ⋯
L(s)  = 1  + (0.631 + 0.775i)3-s + (0.875 + 1.51i)5-s + (−0.202 + 0.979i)9-s + (0.964 + 0.556i)11-s + (−0.133 + 0.0769i)13-s + (−0.622 + 1.63i)15-s + 1.41·17-s − 1.22i·19-s + (0.410 − 0.237i)23-s + (−1.03 + 1.78i)25-s + (−0.886 + 0.461i)27-s + (0.656 + 0.379i)29-s + (1.25 − 0.726i)31-s + (0.177 + 1.09i)33-s − 1.28·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.329 - 0.944i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.329 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.775800350\)
\(L(\frac12)\) \(\approx\) \(2.775800350\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.09 - 1.34i)T \)
7 \( 1 \)
good5 \( 1 + (-1.95 - 3.39i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.19 - 1.84i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.480 - 0.277i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.83T + 17T^{2} \)
19 \( 1 + 5.33iT - 19T^{2} \)
23 \( 1 + (-1.96 + 1.13i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.53 - 2.04i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.00 + 4.04i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.79T + 37T^{2} \)
41 \( 1 + (3.59 + 6.22i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.754 - 1.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.0479iT - 53T^{2} \)
59 \( 1 + (4.45 + 7.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.03 + 3.48i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.587 + 1.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.71iT - 71T^{2} \)
73 \( 1 + 4.07iT - 73T^{2} \)
79 \( 1 + (-1.97 + 3.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.84 - 6.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.42T + 89T^{2} \)
97 \( 1 + (13.9 + 8.07i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.620421776951967867273459556688, −9.032003720573000718957955943669, −7.950988472004299609998777636730, −7.04045528207082297995300862337, −6.50551823916029060526662203145, −5.45146046838084990338665347284, −4.53346613381046903985359112981, −3.37779944571588165910585396139, −2.83203792492312316236954275474, −1.78796362771335283523216435419, 1.11857787356516582090747262251, 1.50657208950268805323975477257, 2.95595253107974319018282504805, 3.95373015927698233990607465800, 5.11544610443997544802094068133, 5.88195039818092733972482664874, 6.54895589681237225910635085860, 7.69576877600684883372776678230, 8.478122907860758858004090752906, 8.803380769397776252265153658575

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