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Properties

Label 2-42e2-63.4-c1-0-14
Degree $2$
Conductor $1764$
Sign $0.169 - 0.985i$
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  + (0.796 + 1.53i)3-s − 2.46·5-s + (−1.73 + 2.45i)9-s + 4.64·11-s + (3.55 − 6.15i)13-s + (−1.96 − 3.78i)15-s + (−2.25 + 3.90i)17-s + (2.16 + 3.74i)19-s + 5.86·23-s + 1.05·25-s + (−5.14 − 0.708i)27-s + (3.48 + 6.04i)29-s + (−3.69 − 6.39i)31-s + (3.70 + 7.14i)33-s + (0.363 + 0.629i)37-s + ⋯
L(s)  = 1  + (0.460 + 0.887i)3-s − 1.10·5-s + (−0.576 + 0.816i)9-s + 1.40·11-s + (0.985 − 1.70i)13-s + (−0.506 − 0.977i)15-s + (−0.547 + 0.948i)17-s + (0.496 + 0.859i)19-s + 1.22·23-s + 0.210·25-s + (−0.990 − 0.136i)27-s + (0.647 + 1.12i)29-s + (−0.662 − 1.14i)31-s + (0.644 + 1.24i)33-s + (0.0597 + 0.103i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.169 - 0.985i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.169 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.745180966\)
\(L(\frac12)\) \(\approx\) \(1.745180966\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.796 - 1.53i)T \)
7 \( 1 \)
good5 \( 1 + 2.46T + 5T^{2} \)
11 \( 1 - 4.64T + 11T^{2} \)
13 \( 1 + (-3.55 + 6.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.25 - 3.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.16 - 3.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.86T + 23T^{2} \)
29 \( 1 + (-3.48 - 6.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.69 + 6.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.363 - 0.629i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.136 + 0.236i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.83 + 3.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.52 - 4.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.56 - 7.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.90 - 11.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.663 - 1.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + (2.16 - 3.74i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.21 - 5.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.742 - 1.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.91 - 8.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.246 + 0.426i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.326636278618629704599424788110, −8.635923276414846500284065808069, −8.111534165550403024214771764866, −7.32250673695448208747266853618, −6.15142539036132334088933818380, −5.34995138917282112848085214554, −4.13433494105519308686685794403, −3.74827279396622858898925146244, −2.95108201143149161807474604699, −1.17705394834355514765414607124, 0.75638376076710610580521878603, 1.89458836593563561878623101743, 3.22068623595515389779986795982, 3.95349982597241118491936101696, 4.82225601993152097736807918350, 6.38086544059736726265245613808, 6.80151907090975578516629092407, 7.39585264716728938431503267097, 8.401855110418572756549881339654, 9.119112651653351755027793605929

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