Properties

Degree $2$
Conductor $1764$
Sign $-0.876 - 0.480i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.629 − 1.26i)2-s + (−1.20 − 1.59i)4-s − 2.32i·5-s + (−2.77 + 0.524i)8-s + (−2.94 − 1.46i)10-s − 3.58i·11-s − 2.93i·13-s + (−1.08 + 3.84i)16-s − 2.32i·17-s + 8.33·19-s + (−3.71 + 2.80i)20-s + (−4.53 − 2.25i)22-s − 1.48i·23-s − 0.414·25-s + (−3.71 − 1.84i)26-s + ⋯
L(s)  = 1  + (0.445 − 0.895i)2-s + (−0.603 − 0.797i)4-s − 1.04i·5-s + (−0.982 + 0.185i)8-s + (−0.931 − 0.463i)10-s − 1.07i·11-s − 0.812i·13-s + (−0.271 + 0.962i)16-s − 0.564i·17-s + 1.91·19-s + (−0.829 + 0.628i)20-s + (−0.966 − 0.480i)22-s − 0.309i·23-s − 0.0828·25-s + (−0.727 − 0.361i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.876 - 0.480i$
Motivic weight: \(1\)
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.876 - 0.480i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.574427471\)
\(L(\frac12)\) \(\approx\) \(1.574427471\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.629 + 1.26i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2.32iT - 5T^{2} \)
11 \( 1 + 3.58iT - 11T^{2} \)
13 \( 1 + 2.93iT - 13T^{2} \)
17 \( 1 + 2.32iT - 17T^{2} \)
19 \( 1 - 8.33T + 19T^{2} \)
23 \( 1 + 1.48iT - 23T^{2} \)
29 \( 1 + 7.86T + 29T^{2} \)
31 \( 1 + 3.45T + 31T^{2} \)
37 \( 1 + 8.24T + 37T^{2} \)
41 \( 1 - 2.89iT - 41T^{2} \)
43 \( 1 - 6.37iT - 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + 8.59T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 2.93iT - 61T^{2} \)
67 \( 1 - 9.02iT - 67T^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 + 11.2iT - 73T^{2} \)
79 \( 1 - 15.3iT - 79T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 + 13.5iT - 89T^{2} \)
97 \( 1 + 5.35iT - 97T^{2} \)
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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

−9.168054185986323628085785300663, −8.255430348505038493432912664416, −7.39193772846249967901715334808, −5.97230173701373003112002441326, −5.38567346992501183379735471215, −4.78376604296081880142229882179, −3.57446334539842956756018566850, −2.95282673667154979803073388281, −1.45729274791901545824670615685, −0.52535952304816477550715255870, 1.95948837231462828078858027507, 3.25205199122049381644273551790, 3.91397213395005154718189124130, 5.02346166929599012848377685350, 5.75044200631289104017080828867, 6.76691891339819315234837276462, 7.26006490427896295169244700706, 7.74512884500019815057825524686, 9.053126642953072670189908623218, 9.487276197472208843178708098662

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