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Properties

Label 2-42e2-84.83-c0-0-1
Degree $2$
Conductor $1764$
Sign $0.860 + 0.508i$
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  − i·2-s − 4-s − 1.84·5-s + i·8-s + 1.84i·10-s + 0.765i·13-s + 16-s + 0.765·17-s + 1.84·20-s + 2.41·25-s + 0.765·26-s − i·32-s − 0.765i·34-s + 1.41·37-s − 1.84i·40-s + 0.765·41-s + ⋯
L(s)  = 1  − i·2-s − 4-s − 1.84·5-s + i·8-s + 1.84i·10-s + 0.765i·13-s + 16-s + 0.765·17-s + 1.84·20-s + 2.41·25-s + 0.765·26-s − i·32-s − 0.765i·34-s + 1.41·37-s − 1.84i·40-s + 0.765·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.860 + 0.508i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 0.860 + 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6624099406\)
\(L(\frac12)\) \(\approx\) \(0.6624099406\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 1.84T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 0.765iT - T^{2} \)
17 \( 1 - 0.765T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 1.41T + T^{2} \)
41 \( 1 - 0.765T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.84iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.84iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.84T + T^{2} \)
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

−9.352145206480679098958886734490, −8.772189353640066310297205445887, −7.82127463875388487402205214139, −7.48103229690311245718116106625, −6.16233727476504975539230726000, −4.89942674519273990336170885420, −4.21614727837055322269356924867, −3.57383524391651644464231861577, −2.62139492364066400511973267529, −1.01789687975060917556622782686, 0.69618160730337280507733034771, 3.11494875182472100248204663555, 3.85349063105782053290624673275, 4.64838935430780894311518710053, 5.49911218432750931644492931431, 6.53439324684329063900248960031, 7.38244563290676307359930398588, 7.943783045695417567617754808633, 8.340617690853894043658166787917, 9.328775289965154341342609305090

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