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Properties

Label 2-42e2-21.11-c2-0-25
Degree $2$
Conductor $1764$
Sign $-0.999 + 0.0348i$
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  + (2.01 − 1.16i)5-s + (−7.70 − 4.44i)11-s + 8.58·13-s + (−17.4 − 10.0i)17-s + (−2.93 − 5.08i)19-s + (15.0 − 8.69i)23-s + (−9.79 + 16.9i)25-s + 19.0i·29-s + (−2.35 + 4.07i)31-s + (−19.8 − 34.4i)37-s + 6.98i·41-s − 35.7·43-s + (−63.5 + 36.6i)47-s + (−40.4 − 23.3i)53-s − 20.7·55-s + ⋯
L(s)  = 1  + (0.403 − 0.232i)5-s + (−0.700 − 0.404i)11-s + 0.660·13-s + (−1.02 − 0.591i)17-s + (−0.154 − 0.267i)19-s + (0.654 − 0.377i)23-s + (−0.391 + 0.678i)25-s + 0.656i·29-s + (−0.0759 + 0.131i)31-s + (−0.537 − 0.930i)37-s + 0.170i·41-s − 0.831·43-s + (−1.35 + 0.780i)47-s + (−0.762 − 0.440i)53-s − 0.376·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0348i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.999 + 0.0348i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ -0.999 + 0.0348i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3152405797\)
\(L(\frac12)\) \(\approx\) \(0.3152405797\)
\(L(2)\) 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 \( 1 + (-2.01 + 1.16i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (7.70 + 4.44i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 8.58T + 169T^{2} \)
17 \( 1 + (17.4 + 10.0i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (2.93 + 5.08i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-15.0 + 8.69i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 19.0iT - 841T^{2} \)
31 \( 1 + (2.35 - 4.07i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (19.8 + 34.4i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 6.98iT - 1.68e3T^{2} \)
43 \( 1 + 35.7T + 1.84e3T^{2} \)
47 \( 1 + (63.5 - 36.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (40.4 + 23.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-20.8 - 12.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-36.6 - 63.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-16.0 + 27.7i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 114. iT - 5.04e3T^{2} \)
73 \( 1 + (20.6 - 35.7i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-13.3 - 23.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 136. iT - 6.88e3T^{2} \)
89 \( 1 + (126. - 72.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 65.9T + 9.40e3T^{2} \)
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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

−8.827305547159466666169251229588, −8.033659788896690973354462184370, −7.06172292755577630665660999011, −6.34060548780462038452130552420, −5.38171424663048927684156863453, −4.75705149129616600856149123045, −3.56782421049588100405713766325, −2.61259911471477943540933638490, −1.47181403067582407383409041327, −0.07753935284422485687058335217, 1.59023274522379661862669523248, 2.51443209701026312017147226552, 3.62396493566511424145560051114, 4.59647976200405252308459130009, 5.49524487427662110681150170223, 6.38149960124830469740311274903, 6.97063965162422173912797204063, 8.129580755060548134369733413021, 8.539248510718633521982563206806, 9.673470942050693461903058505006

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