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

\(L(1)\) \(\approx\) \(1.189606397\)
\(L(\frac12)\) \(\approx\) \(1.189606397\)
\(L(\frac{3}{2})\) 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

−9.415031666698429248866399554469, −8.569997210141214568932619407132, −7.57987629864807306330883108038, −7.29476750798436871974007871446, −6.32228063502552642530878523169, −5.56114543391741238765336201377, −4.74156242101763423484470598179, −3.35769054750328220706337392777, −2.33095361418293132161863258608, −0.835315931188436297924140425206, 0.872182168283802581494826426066, 1.87817368842723006370202751786, 3.10004910286515755511790684133, 4.06146484947744979374346291766, 4.88980461809374284139112356957, 5.99528243713056950803383651794, 7.09103375691693862996366538239, 7.78576811091325290067089999008, 8.666071664420291904168486894168, 9.185304582467199686391351913459

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