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Properties

Label 2-42e2-63.16-c1-0-3
Degree $2$
Conductor $1764$
Sign $0.607 - 0.794i$
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 − 0.239·5-s + (−0.619 + 2.93i)9-s − 5.12·11-s + (−2.44 − 4.23i)13-s + (0.260 + 0.321i)15-s + (1.85 + 3.20i)17-s + (1.83 − 3.16i)19-s + 7.42·23-s − 4.94·25-s + (4.62 − 2.36i)27-s + (−1.73 + 3.00i)29-s + (−0.358 + 0.621i)31-s + (5.59 + 6.89i)33-s + (−2.30 + 3.98i)37-s + ⋯
L(s)  = 1  + (−0.629 − 0.776i)3-s − 0.106·5-s + (−0.206 + 0.978i)9-s − 1.54·11-s + (−0.677 − 1.17i)13-s + (0.0673 + 0.0830i)15-s + (0.449 + 0.777i)17-s + (0.419 − 0.727i)19-s + 1.54·23-s − 0.988·25-s + (0.890 − 0.455i)27-s + (−0.321 + 0.557i)29-s + (−0.0644 + 0.111i)31-s + (0.973 + 1.20i)33-s + (−0.378 + 0.655i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.607 - 0.794i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.607 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6791095931\)
\(L(\frac12)\) \(\approx\) \(0.6791095931\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.239T + 5T^{2} \)
11 \( 1 + 5.12T + 11T^{2} \)
13 \( 1 + (2.44 + 4.23i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.85 - 3.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.83 + 3.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.42T + 23T^{2} \)
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 \( 1 + (2.30 - 3.98i)T + (-18.5 - 32.0i)T^{2} \)
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 \( 1 + (-0.471 - 0.816i)T + (-26.5 + 45.8i)T^{2} \)
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 \( 1 + (1.83 + 3.16i)T + (-36.5 + 63.2i)T^{2} \)
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 \( 1 + (3.74 - 6.48i)T + (-44.5 - 77.0i)T^{2} \)
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

−9.526520689155298069082531649395, −8.222899400067016341275208014390, −7.84110165429707193855912870337, −7.14428228431155965676925407541, −6.18623268560167116915190948657, −5.20576468987365390538091285409, −4.98115747867143171383974188038, −3.22659559251360490006957659093, −2.45060064034879559194722941487, −1.01150653871650530986157694161, 0.32627477053542778284580821855, 2.20070132380799128595824935530, 3.34774329532091358999179053052, 4.28653190837483950952335147241, 5.28679515631676867374576947008, 5.54046151016659729786977869664, 6.90570425482377561937186463906, 7.44223467834495514372472397217, 8.508460465517126363986737344751, 9.414230367742922149717260532849

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