Properties

Degree $2$
Conductor $1764$
Sign $-0.654 + 0.755i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + 2i·4-s + 1.73i·5-s + (2 − 2i)8-s + (1.73 − 1.73i)10-s i·11-s − 3.46i·13-s − 4·16-s − 1.73i·17-s − 5.19·19-s − 3.46·20-s + (−1 + i)22-s + i·23-s + 2.00·25-s + (−3.46 + 3.46i)26-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + i·4-s + 0.774i·5-s + (0.707 − 0.707i)8-s + (0.547 − 0.547i)10-s − 0.301i·11-s − 0.960i·13-s − 16-s − 0.420i·17-s − 1.19·19-s − 0.774·20-s + (−0.213 + 0.213i)22-s + 0.208i·23-s + 0.400·25-s + (−0.679 + 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.654 + 0.755i$
Motivic weight: \(1\)
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6467214703\)
\(L(\frac12)\) \(\approx\) \(0.6467214703\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 1.73iT - 5T^{2} \)
11 \( 1 + iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + 1.73iT - 17T^{2} \)
19 \( 1 + 5.19T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 1.73T + 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 8.66T + 47T^{2} \)
53 \( 1 - T + 53T^{2} \)
59 \( 1 - 5.19T + 59T^{2} \)
61 \( 1 + 5.19iT - 61T^{2} \)
67 \( 1 - 3iT - 67T^{2} \)
71 \( 1 + 14iT - 71T^{2} \)
73 \( 1 + 8.66iT - 73T^{2} \)
79 \( 1 + 9iT - 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + 15.5iT - 89T^{2} \)
97 \( 1 - 17.3iT - 97T^{2} \)
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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

−9.057397895034556557974848523313, −8.329821646066909680617108574602, −7.56311283060353037257645370358, −6.84829286702993494992917504166, −5.92022651526108284663064294371, −4.71189647367414077419706756442, −3.58333101498185773125205387877, −2.91149061571937279277692471832, −1.89323360354094045278905221403, −0.32154438462153693801956957110, 1.26575532208995374702107014886, 2.27290889800607083619168144811, 4.08393409131228156063404608575, 4.75662625396817578646449208347, 5.68456782931109957100422552617, 6.54275645516538734442746771850, 7.18026369553974865960577019010, 8.254051450185153769513668680817, 8.644089711526522250106088226725, 9.442876979820175082931072656269

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