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Properties

Label 2-42e2-63.5-c1-0-23
Degree $2$
Conductor $1764$
Sign $0.999 + 0.00687i$
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  + (1.36 − 1.06i)3-s + (1.48 + 2.57i)5-s + (0.727 − 2.91i)9-s + (−4.09 − 2.36i)11-s + (3.54 + 2.04i)13-s + (4.76 + 1.92i)15-s + (0.835 + 1.44i)17-s + (4.25 + 2.45i)19-s + (4.25 − 2.45i)23-s + (−1.91 + 3.30i)25-s + (−2.11 − 4.74i)27-s + (0.238 − 0.137i)29-s − 1.60i·31-s + (−8.10 + 1.13i)33-s + (−1.69 + 2.93i)37-s + ⋯
L(s)  = 1  + (0.788 − 0.615i)3-s + (0.664 + 1.15i)5-s + (0.242 − 0.970i)9-s + (−1.23 − 0.712i)11-s + (0.981 + 0.566i)13-s + (1.23 + 0.497i)15-s + (0.202 + 0.350i)17-s + (0.975 + 0.563i)19-s + (0.886 − 0.511i)23-s + (−0.382 + 0.661i)25-s + (−0.406 − 0.913i)27-s + (0.0442 − 0.0255i)29-s − 0.287i·31-s + (−1.41 + 0.198i)33-s + (−0.278 + 0.483i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.00687i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.999 + 0.00687i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.666801146\)
\(L(\frac12)\) \(\approx\) \(2.666801146\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 + 1.06i)T \)
7 \( 1 \)
good5 \( 1 + (-1.48 - 2.57i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.09 + 2.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.54 - 2.04i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.835 - 1.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.25 - 2.45i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.25 + 2.45i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.238 + 0.137i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.60iT - 31T^{2} \)
37 \( 1 + (1.69 - 2.93i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.55 + 6.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.22 - 9.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + (-0.707 + 0.408i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.74T + 59T^{2} \)
61 \( 1 - 7.20iT - 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (13.6 - 7.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + (-4.03 - 6.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.60 - 7.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.00 - 4.04i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.207636666406059436764492511230, −8.460449677865038168921081900157, −7.68117022786758709525139867339, −6.98781343601526173011575427073, −6.15287693321306573244008380986, −5.57641699595721885028611134350, −4.00082077041218616934433186263, −3.02302489103364329286364377569, −2.51374833289180425685573476747, −1.22328826515031697890453888056, 1.12431139556437790455427650175, 2.37511671936585085002153681399, 3.28355867288261277543048917167, 4.44772434651090858240210398546, 5.24491785300362894858397704440, 5.61704376287365739406289581602, 7.23283842008438403569818425512, 7.81816488390214331390427089226, 8.792699583628970124926317722086, 9.136775085776268826394055044371

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