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Properties

Label 2-42e2-21.5-c1-0-3
Degree $2$
Conductor $1764$
Sign $-0.536 - 0.843i$
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  + (−0.923 + 1.60i)5-s + (1.73 − i)11-s + 4.46i·13-s + (1.14 + 1.98i)17-s + (1.32 + 0.765i)19-s + (−7.64 − 4.41i)23-s + (0.792 + 1.37i)25-s − 1.17i·29-s + (5.07 − 2.93i)31-s + (−4.12 + 7.13i)37-s − 11.8·41-s + 1.17·43-s + (−4.01 + 6.94i)47-s + (3.25 − 1.87i)53-s + 3.69i·55-s + ⋯
L(s)  = 1  + (−0.413 + 0.715i)5-s + (0.522 − 0.301i)11-s + 1.23i·13-s + (0.278 + 0.482i)17-s + (0.304 + 0.175i)19-s + (−1.59 − 0.920i)23-s + (0.158 + 0.274i)25-s − 0.217i·29-s + (0.911 − 0.526i)31-s + (−0.677 + 1.17i)37-s − 1.85·41-s + 0.178·43-s + (−0.585 + 1.01i)47-s + (0.446 − 0.258i)53-s + 0.498i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.536 - 0.843i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.536 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.105738710\)
\(L(\frac12)\) \(\approx\) \(1.105738710\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.923 - 1.60i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.73 + i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.46iT - 13T^{2} \)
17 \( 1 + (-1.14 - 1.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.32 - 0.765i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.64 + 4.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.17iT - 29T^{2} \)
31 \( 1 + (-5.07 + 2.93i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.12 - 7.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 1.17T + 43T^{2} \)
47 \( 1 + (4.01 - 6.94i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.25 + 1.87i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.90 - 8.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.6 - 6.15i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.24 + 10.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.3iT - 71T^{2} \)
73 \( 1 + (2.37 - 1.37i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.65 - 9.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + (7.23 - 12.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.74iT - 97T^{2} \)
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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

−9.738347565282756057494085248417, −8.586976562755797136363183146436, −8.135229397439498689843705615272, −6.97730613597887120200170782677, −6.57296084381934377444819855847, −5.65687986951034493633722700037, −4.39060526208178284603667751368, −3.78008627437158129268624412562, −2.71598038501601336731984049231, −1.50373719802504994164737598278, 0.42135657984947200178975323908, 1.73590163173177323597998725148, 3.14739615902203379525399248393, 3.98340513371999296347542684779, 5.00582256943184616601804127869, 5.61275484997873557604800173480, 6.70784277282993078108634051903, 7.56588578630655932849246490018, 8.283057038741251138491261271884, 8.883935605016827958159123823480

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