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Properties

Label 2-42e2-63.25-c1-0-13
Degree $2$
Conductor $1764$
Sign $0.982 - 0.185i$
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  + (−1.73 + 0.0789i)3-s + (1.23 − 2.13i)5-s + (2.98 − 0.273i)9-s + (−2.32 − 4.02i)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 + (−2.93 + 5.08i)23-s + (−0.527 − 0.912i)25-s + (−5.14 + 0.708i)27-s + (3.48 − 6.04i)29-s + 7.38·31-s + (4.33 + 6.78i)33-s + (0.363 + 0.629i)37-s + ⋯
L(s)  = 1  + (−0.998 + 0.0455i)3-s + (0.550 − 0.952i)5-s + (0.995 − 0.0910i)9-s + (−0.700 − 1.21i)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 + (−0.611 + 1.05i)23-s + (−0.105 − 0.182i)25-s + (−0.990 + 0.136i)27-s + (0.647 − 1.12i)29-s + 1.32·31-s + (0.755 + 1.18i)33-s + (0.0597 + 0.103i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.982 - 0.185i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.982 - 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.314706197\)
\(L(\frac12)\) \(\approx\) \(1.314706197\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 0.0789i)T \)
7 \( 1 \)
good5 \( 1 + (-1.23 + 2.13i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.32 + 4.02i)T + (-5.5 + 9.52i)T^{2} \)
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 \( 1 + (2.93 - 5.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.48 + 6.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.38T + 31T^{2} \)
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 \( 1 + 3.67T + 47T^{2} \)
53 \( 1 + (2.52 - 4.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.13T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + 1.32T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + (2.16 - 3.74i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 6.43T + 79T^{2} \)
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

−9.390152886405638094853162483204, −8.543718862076785405517055699640, −7.902050346661306411030592728650, −6.53932738947351215241936674688, −6.06044752094360033331892972345, −5.41600886161736033673530165024, −4.44577754145213558161429741366, −3.69627280775408684921593904159, −1.91123134719408484562362337730, −1.00584715777561114315652279556, 0.72151004184315763087530136427, 2.31507835480287492552750385439, 3.13763341987866866710882619346, 4.64290608845992127948350487556, 5.17006830660022449466311797100, 6.20291463544325608003804986095, 6.71926710908702454194039123005, 7.48932523917413223118088919680, 8.356773825882436121787724102809, 9.638213628032656037114831356673

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