Properties

Degree $2$
Conductor $1764$
Sign $-0.386 + 0.922i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·13-s + (−0.5 − 0.866i)19-s + (2.5 − 4.33i)25-s + (5.5 − 9.52i)31-s + (−5.5 − 9.52i)37-s − 13·43-s + (7 + 12.1i)61-s + (−2.5 + 4.33i)67-s + (8.5 − 14.7i)73-s + (−8.5 − 14.7i)79-s − 14·97-s + (−6.5 − 11.2i)103-s + (9.5 − 16.4i)109-s + ⋯
L(s)  = 1  − 1.38·13-s + (−0.114 − 0.198i)19-s + (0.5 − 0.866i)25-s + (0.987 − 1.71i)31-s + (−0.904 − 1.56i)37-s − 1.98·43-s + (0.896 + 1.55i)61-s + (−0.305 + 0.529i)67-s + (0.994 − 1.72i)73-s + (−0.956 − 1.65i)79-s − 1.42·97-s + (−0.640 − 1.10i)103-s + (0.909 − 1.57i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.386 + 0.922i$
Motivic weight: \(1\)
Character: $\chi_{1764} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9123965823\)
\(L(\frac12)\) \(\approx\) \(0.9123965823\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-5.5 + 9.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 13T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-8.5 + 14.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−9.065661880740095282906361013185, −8.255530357163612598956641288302, −7.44085467768578459224412039613, −6.75045881725246026399194981917, −5.79784314409085780556131012788, −4.89580628301305627995229613751, −4.14643365204930599716803810902, −2.90917606892586871300613969247, −2.03418938979461441867474811686, −0.33433785379799074607024525153, 1.43413597535384233558640062636, 2.68442837366993884113119511113, 3.56340001883532626000430154013, 4.88594222999733455436622366586, 5.19829198354000330994047648204, 6.61471505043786677071393997187, 6.99228793999434855221368045232, 8.080852764483335655557793315926, 8.647372999198639884868872893249, 9.733484805010312995757072075307

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