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Properties

Label 2-42e2-9.4-c1-0-12
Degree $2$
Conductor $1764$
Sign $0.140 - 0.990i$
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  + (1.71 + 0.272i)3-s + (0.119 − 0.207i)5-s + (2.85 + 0.931i)9-s + (2.56 + 4.43i)11-s + (−2.44 + 4.23i)13-s + (0.260 − 0.321i)15-s − 3.70·17-s − 3.66·19-s + (−3.71 + 6.42i)23-s + (2.47 + 4.28i)25-s + (4.62 + 2.36i)27-s + (−1.73 − 3.00i)29-s + (−0.358 + 0.621i)31-s + (3.17 + 8.28i)33-s + 4.60·37-s + ⋯
L(s)  = 1  + (0.987 + 0.157i)3-s + (0.0534 − 0.0926i)5-s + (0.950 + 0.310i)9-s + (0.772 + 1.33i)11-s + (−0.677 + 1.17i)13-s + (0.0673 − 0.0830i)15-s − 0.898·17-s − 0.839·19-s + (−0.773 + 1.34i)23-s + (0.494 + 0.856i)25-s + (0.890 + 0.455i)27-s + (−0.321 − 0.557i)29-s + (−0.0644 + 0.111i)31-s + (0.552 + 1.44i)33-s + 0.756·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.140 - 0.990i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.140 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.244473397\)
\(L(\frac12)\) \(\approx\) \(2.244473397\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.272i)T \)
7 \( 1 \)
good5 \( 1 + (-0.119 + 0.207i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.56 - 4.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.44 - 4.23i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.70T + 17T^{2} \)
19 \( 1 + 3.66T + 19T^{2} \)
23 \( 1 + (3.71 - 6.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.73 + 3.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.358 - 0.621i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.60T + 37T^{2} \)
41 \( 1 + (-2.80 + 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.24 + 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.16 - 3.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.942T + 53T^{2} \)
59 \( 1 + (-3.78 + 6.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.75 - 4.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.330 + 0.571i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 - 3.66T + 73T^{2} \)
79 \( 1 + (-3.11 - 5.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.85 - 8.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.48T + 89T^{2} \)
97 \( 1 + (8.57 + 14.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.423035084649114526932334616913, −8.908136700471810768468211366553, −7.901957330329553626563187086985, −7.08128039353318888544255569866, −6.64928418357528421919849784257, −5.19394209981634512370633615092, −4.26150615003033586097701064175, −3.81511755197023790480399028953, −2.25506716959470395903380307495, −1.79663080144774905293221094850, 0.71687220000516924054773501633, 2.24984536710322698267771634158, 3.00620900138416945465463444778, 3.97419985136907493367635670964, 4.83499915051380659819745496431, 6.22484627577271810285491863008, 6.59650265739066379166333315535, 7.81209842223105801978990527155, 8.367713005037320592956038453565, 8.912698392142894322547493177589

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