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Properties

Label 2-42e2-7.5-c2-0-27
Degree $2$
Conductor $1764$
Sign $-0.982 + 0.188i$
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  + (−3.68 − 2.12i)5-s + (9.48 + 16.4i)11-s − 4.46i·13-s + (−11.0 + 6.38i)17-s + (−8.11 − 4.68i)19-s + (8.53 − 14.7i)23-s + (−3.44 − 5.97i)25-s + 20.8·29-s + (−13.5 + 7.80i)31-s + (−22.8 + 39.5i)37-s − 63.3i·41-s + 2.20·43-s + (43.8 + 25.3i)47-s + (−18.9 − 32.8i)53-s − 80.7i·55-s + ⋯
L(s)  = 1  + (−0.736 − 0.425i)5-s + (0.862 + 1.49i)11-s − 0.343i·13-s + (−0.650 + 0.375i)17-s + (−0.427 − 0.246i)19-s + (0.371 − 0.642i)23-s + (−0.137 − 0.239i)25-s + 0.720·29-s + (−0.435 + 0.251i)31-s + (−0.617 + 1.06i)37-s − 1.54i·41-s + 0.0511·43-s + (0.932 + 0.538i)47-s + (−0.358 − 0.620i)53-s − 1.46i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.982 + 0.188i$
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: \((2,\ 1764,\ (\ :1),\ -0.982 + 0.188i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1551322804\)
\(L(\frac12)\) \(\approx\) \(0.1551322804\)
\(L(2)\) 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 \( 1 + (3.68 + 2.12i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-9.48 - 16.4i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 4.46iT - 169T^{2} \)
17 \( 1 + (11.0 - 6.38i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (8.11 + 4.68i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-8.53 + 14.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 20.8T + 841T^{2} \)
31 \( 1 + (13.5 - 7.80i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (22.8 - 39.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 63.3iT - 1.68e3T^{2} \)
43 \( 1 - 2.20T + 1.84e3T^{2} \)
47 \( 1 + (-43.8 - 25.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (18.9 + 32.8i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-58.5 + 33.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (54.8 + 31.6i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (13.6 + 23.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 15.1T + 5.04e3T^{2} \)
73 \( 1 + (72.5 - 41.9i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (66.5 - 115. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 109. iT - 6.88e3T^{2} \)
89 \( 1 + (69.2 + 39.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 119. iT - 9.40e3T^{2} \)
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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

−8.672549392825059023606304730907, −8.077543553937740853820949487847, −7.01023440622638385433807107007, −6.63200204462255443929866788448, −5.32690868047995398300973746252, −4.41179416377436580713290784091, −3.97775930068955399478856154188, −2.58596179799123302534568003204, −1.45574512381874933247843699707, −0.04309320904890919743821691544, 1.29606753567595151877026186505, 2.76916266984476807166446783757, 3.64375437709864366485059177976, 4.31929035166449411485779985600, 5.55949478517943062565878466668, 6.34379024864808295918051232243, 7.11005762166157523398953195465, 7.88446232801489827518260615324, 8.819568891350730463193772071689, 9.199996514843592205675445934773

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