Properties

Label 2-42e2-28.27-c1-0-12
Degree $2$
Conductor $1764$
Sign $-0.889 + 0.456i$
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  + (−0.639 + 1.26i)2-s + (−1.18 − 1.61i)4-s + 3.10i·5-s + (2.79 − 0.457i)8-s + (−3.91 − 1.98i)10-s + 5.34i·11-s + 3.92i·13-s + (−1.20 + 3.81i)16-s − 5.68i·17-s + 0.170·19-s + (5.01 − 3.66i)20-s + (−6.74 − 3.41i)22-s + 6.13i·23-s − 4.63·25-s + (−4.95 − 2.51i)26-s + ⋯
L(s)  = 1  + (−0.452 + 0.891i)2-s + (−0.590 − 0.806i)4-s + 1.38i·5-s + (0.986 − 0.161i)8-s + (−1.23 − 0.628i)10-s + 1.61i·11-s + 1.08i·13-s + (−0.302 + 0.953i)16-s − 1.37i·17-s + 0.0391·19-s + (1.12 − 0.819i)20-s + (−1.43 − 0.729i)22-s + 1.27i·23-s − 0.927·25-s + (−0.971 − 0.492i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.889 + 0.456i$
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: \((2,\ 1764,\ (\ :1/2),\ -0.889 + 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9502550263\)
\(L(\frac12)\) \(\approx\) \(0.9502550263\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.639 - 1.26i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 3.10iT - 5T^{2} \)
11 \( 1 - 5.34iT - 11T^{2} \)
13 \( 1 - 3.92iT - 13T^{2} \)
17 \( 1 + 5.68iT - 17T^{2} \)
19 \( 1 - 0.170T + 19T^{2} \)
23 \( 1 - 6.13iT - 23T^{2} \)
29 \( 1 - 1.96T + 29T^{2} \)
31 \( 1 + 1.53T + 31T^{2} \)
37 \( 1 - 8.93T + 37T^{2} \)
41 \( 1 - 5.84iT - 41T^{2} \)
43 \( 1 + 2.38iT - 43T^{2} \)
47 \( 1 + 2.29T + 47T^{2} \)
53 \( 1 - 1.19T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 + 6.58iT - 61T^{2} \)
67 \( 1 - 9.64iT - 67T^{2} \)
71 \( 1 - 2.04iT - 71T^{2} \)
73 \( 1 - 10.2iT - 73T^{2} \)
79 \( 1 + 14.3iT - 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + 5.81iT - 89T^{2} \)
97 \( 1 - 2.32iT - 97T^{2} \)
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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

−9.694097059171403354309076669868, −9.134790361674430975529569772020, −7.84166242089118338507787408463, −7.18481606236496596540857002154, −6.90125815957917996788772152924, −6.01624098095063396466672955025, −4.91530389894337599961791695415, −4.17290746293550740182292076204, −2.80572693948331517014110401689, −1.68641939867601166152092788370, 0.45686335943390899519367271316, 1.29142799974113542859289337937, 2.69797152942220852527873877754, 3.68330149027670548108497839524, 4.53016849401349525679900082282, 5.46573036065287525247540254129, 6.25613654723357950091812176889, 7.85121244052986973727392501511, 8.298198710616631694548670916922, 8.774731492329438341625065108862

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