Properties

Label 2-42e2-84.59-c0-0-4
Degree $2$
Conductor $1764$
Sign $0.524 + 0.851i$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (0.707 − 1.22i)11-s + (0.500 + 0.866i)16-s + (−1 + 0.999i)22-s + (−0.707 − 1.22i)23-s + (0.5 − 0.866i)25-s + 1.41i·29-s + (−0.258 − 0.965i)32-s − 2i·43-s + (1.22 − 0.707i)44-s + (0.366 + 1.36i)46-s + (−0.707 + 0.707i)50-s + (1.22 + 0.707i)53-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (0.707 − 1.22i)11-s + (0.500 + 0.866i)16-s + (−1 + 0.999i)22-s + (−0.707 − 1.22i)23-s + (0.5 − 0.866i)25-s + 1.41i·29-s + (−0.258 − 0.965i)32-s − 2i·43-s + (1.22 − 0.707i)44-s + (0.366 + 1.36i)46-s + (−0.707 + 0.707i)50-s + (1.22 + 0.707i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.524 + 0.851i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 0.524 + 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7387226811\)
\(L(\frac12)\) \(\approx\) \(0.7387226811\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 2iT - T^{2} \)
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 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.124554499606032285548011410478, −8.701477085016473379242981941355, −8.071517023645916358802118712325, −6.98284805552286638866097346565, −6.42735454756639515792821725609, −5.51529773668200935381352613389, −4.12036402573428287972363976573, −3.25031189567312950370319149255, −2.20782116348550708129413046027, −0.843969401835914170350792639920, 1.38649625387249975259853539489, 2.35958524229136551877381810066, 3.68610370810542115210532468104, 4.81354011765740171604957106611, 5.82354279982466946423044187600, 6.61781071881858760903003647943, 7.38425425483051467547392341397, 7.967898959140578750543394773075, 8.927167732103795973802557764971, 9.738960871949649149118001589949

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