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Properties

Label 2-42e2-84.47-c0-0-9
Degree $2$
Conductor $1764$
Sign $0.974 - 0.224i$
Analytic cond. $0.880350$
Root an. cond. $0.938270$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.923 − 1.60i)5-s + 0.999i·8-s + (1.60 − 0.923i)10-s + 0.765i·13-s + (−0.5 + 0.866i)16-s + (−0.382 − 0.662i)17-s + 1.84·20-s + (−1.20 − 2.09i)25-s + (−0.382 + 0.662i)26-s + (−0.866 + 0.499i)32-s − 0.765i·34-s + (−0.707 + 1.22i)37-s + (1.60 + 0.923i)40-s + 0.765·41-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.923 − 1.60i)5-s + 0.999i·8-s + (1.60 − 0.923i)10-s + 0.765i·13-s + (−0.5 + 0.866i)16-s + (−0.382 − 0.662i)17-s + 1.84·20-s + (−1.20 − 2.09i)25-s + (−0.382 + 0.662i)26-s + (−0.866 + 0.499i)32-s − 0.765i·34-s + (−0.707 + 1.22i)37-s + (1.60 + 0.923i)40-s + 0.765·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.974 - 0.224i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 0.974 - 0.224i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.126541617\)
\(L(\frac12)\) \(\approx\) \(2.126541617\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - 0.765iT - T^{2} \)
17 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - 0.765T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.60 - 0.923i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.84iT - T^{2} \)
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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

−9.247921968302132748115958352971, −8.750531279254604723505556781936, −7.968513772560903293682508439818, −6.91785698398769212827721805852, −6.16004758784143046034202239441, −5.32157802493041326130565347426, −4.76824448801656204420283114932, −4.01464917450545036754445204715, −2.58874163696641962113979564878, −1.54574740444573382898579389523, 1.76214967257629880486954558777, 2.66187345004498787139801367263, 3.34377010565217061031362112949, 4.36401294119883361986638010687, 5.76032570795371438607034991162, 5.89476955870057440798421982459, 6.93637303082961526894799416395, 7.48702428855222534343240557190, 8.914926578966166700924130948379, 9.814495354905709674096300764599

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