Properties

Label 2-42e2-28.27-c1-0-3
Degree $2$
Conductor $1764$
Sign $-0.156 - 0.987i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 2.00·4-s − 4.46i·5-s − 2.82·8-s + 6.30i·10-s + 5.99i·13-s + 4.00·16-s + 4.90i·17-s − 8.92i·20-s − 14.8·25-s − 8.47i·26-s − 4.24·29-s − 5.65·32-s − 6.94i·34-s − 9.89·37-s + ⋯
L(s)  = 1  − 1.00·2-s + 1.00·4-s − 1.99i·5-s − 1.00·8-s + 1.99i·10-s + 1.66i·13-s + 1.00·16-s + 1.19i·17-s − 1.99i·20-s − 2.97·25-s − 1.66i·26-s − 0.787·29-s − 1.00·32-s − 1.19i·34-s − 1.62·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.156 - 0.987i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.156 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3244458205\)
\(L(\frac12)\) \(\approx\) \(0.3244458205\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 4.46iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5.99iT - 13T^{2} \)
17 \( 1 - 4.90iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 9.89T + 37T^{2} \)
41 \( 1 - 3.56iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 14T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 7.25iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 11.6iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 3.11iT - 89T^{2} \)
97 \( 1 + 13.5iT - 97T^{2} \)
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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

−9.356554060367622368000253942561, −8.711258880341893226247718626496, −8.305912971010849828366233237750, −7.37124960891758479558074730343, −6.37674509318586288949630974728, −5.56601353717950790132646035570, −4.58421610699766338929106211156, −3.73460939411724930315780139620, −1.92154934375493379713119539535, −1.37804314961950780359929871711, 0.16461435941883506991208667661, 2.03410388278795894468886233669, 3.02624746462146134096714794570, 3.42063437193610336860236007917, 5.32604597470365176931109063674, 6.15958295492554414862253100012, 6.92909844142046773348769744407, 7.52124561352620427516376049835, 8.069673213702405694465189707282, 9.260479145174871549266686026321

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