Properties

Label 2-42e2-84.83-c0-0-4
Degree $2$
Conductor $1764$
Sign $0.192 - 0.981i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.192 - 0.981i$
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: \((2,\ 1764,\ (\ :0),\ 0.192 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.387183319\)
\(L(\frac12)\) \(\approx\) \(1.387183319\)
\(L(1)\) 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

−9.449228039087642369900110934950, −8.995531086374524324565891613718, −8.142709213880847615094332474154, −6.95113624248823697267981143064, −6.49458202630108108758718598442, −5.75842209397954328063701618222, −5.03879713414506283038004542888, −4.14767984974902007584685434526, −2.67107320136561336442808555938, −1.52861518672177461537377006355, 1.28862775764033900154475398541, 2.30140401196221761589119658538, 2.98548705900688903005765360359, 4.32682505765461180980622422072, 5.26079863255650564321353082630, 5.85524836894980512040236051750, 6.74090158127873025714112256832, 8.040133557904187252417891712685, 8.860008671871103150296670583598, 9.550886210795606231658646905120

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