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Properties

Label 2-42e2-7.5-c2-0-12
Degree $2$
Conductor $1764$
Sign $-0.0633 - 0.997i$
Analytic cond. $48.0655$
Root an. cond. $6.93293$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6 + 3.46i)5-s + (9 + 15.5i)11-s + 20.7i·13-s + (12 − 6.92i)17-s + (18 + 10.3i)19-s + (9 − 15.5i)23-s + (11.5 + 19.9i)25-s − 18·29-s + (−36 + 20.7i)31-s + (−5 + 8.66i)37-s − 55.4i·41-s − 38·43-s + (24 + 13.8i)47-s + (9 + 15.5i)53-s + 124. i·55-s + ⋯
L(s)  = 1  + (1.20 + 0.692i)5-s + (0.818 + 1.41i)11-s + 1.59i·13-s + (0.705 − 0.407i)17-s + (0.947 + 0.546i)19-s + (0.391 − 0.677i)23-s + (0.460 + 0.796i)25-s − 0.620·29-s + (−1.16 + 0.670i)31-s + (−0.135 + 0.234i)37-s − 1.35i·41-s − 0.883·43-s + (0.510 + 0.294i)47-s + (0.169 + 0.294i)53-s + 2.26i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0633 - 0.997i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ -0.0633 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.717074715\)
\(L(\frac12)\) \(\approx\) \(2.717074715\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-6 - 3.46i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-9 - 15.5i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 20.7iT - 169T^{2} \)
17 \( 1 + (-12 + 6.92i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-18 - 10.3i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-9 + 15.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 18T + 841T^{2} \)
31 \( 1 + (36 - 20.7i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (5 - 8.66i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 55.4iT - 1.68e3T^{2} \)
43 \( 1 + 38T + 1.84e3T^{2} \)
47 \( 1 + (-24 - 13.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-9 - 15.5i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-6 + 3.46i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (18 + 10.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (13 + 22.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 18T + 5.04e3T^{2} \)
73 \( 1 + (36 - 20.7i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 34.6iT - 6.88e3T^{2} \)
89 \( 1 + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 166. iT - 9.40e3T^{2} \)
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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

−9.418424272841323764227715369530, −8.864956152914921626841809432999, −7.31476699166468149659023983140, −7.04245827758474868514760237785, −6.20170086915726709040877584897, −5.32658404359203789757040842656, −4.38571179292568221083089352973, −3.35108834369640619821574707017, −2.09797031734048593897342966574, −1.54850006654660021821734743264, 0.72640893182919724231340655995, 1.52095303138891607393038343433, 2.96100218347333540272185247394, 3.67998996618958484578454607537, 5.20403331526024162053739678405, 5.59301472513993081906931542215, 6.20431932661369799939291462895, 7.42432369104263778864859277780, 8.226584609758472056381064421066, 9.036855902407684782419273321089

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