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Properties

Label 2-42e2-21.2-c2-0-11
Degree $2$
Conductor $1764$
Sign $0.163 - 0.986i$
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  + (8.27 + 4.77i)5-s + (8.03 − 4.63i)11-s + 4.24·13-s + (−13.4 + 7.77i)17-s + (−17.4 + 30.2i)19-s + (15.3 + 8.87i)23-s + (33.1 + 57.4i)25-s − 26.2i·29-s + (−16.0 − 27.7i)31-s + (−27.6 + 47.9i)37-s + 38.4i·41-s + 29.3·43-s + (59.0 + 34.1i)47-s + (−0.943 + 0.544i)53-s + 88.6·55-s + ⋯
L(s)  = 1  + (1.65 + 0.955i)5-s + (0.730 − 0.421i)11-s + 0.326·13-s + (−0.792 + 0.457i)17-s + (−0.918 + 1.59i)19-s + (0.668 + 0.386i)23-s + (1.32 + 2.29i)25-s − 0.904i·29-s + (−0.517 − 0.895i)31-s + (−0.747 + 1.29i)37-s + 0.937i·41-s + 0.682·43-s + (1.25 + 0.725i)47-s + (−0.0177 + 0.0102i)53-s + 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.163 - 0.986i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1745, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ 0.163 - 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.776312738\)
\(L(\frac12)\) \(\approx\) \(2.776312738\)
\(L(2)\) 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 \( 1 + (-8.27 - 4.77i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-8.03 + 4.63i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 4.24T + 169T^{2} \)
17 \( 1 + (13.4 - 7.77i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (17.4 - 30.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-15.3 - 8.87i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 26.2iT - 841T^{2} \)
31 \( 1 + (16.0 + 27.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (27.6 - 47.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 38.4iT - 1.68e3T^{2} \)
43 \( 1 - 29.3T + 1.84e3T^{2} \)
47 \( 1 + (-59.0 - 34.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (0.943 - 0.544i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (59.0 - 34.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-23.8 + 41.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-4.32 - 7.49i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 95.7iT - 5.04e3T^{2} \)
73 \( 1 + (-45.0 - 77.9i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-74.3 + 128. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 88.8iT - 6.88e3T^{2} \)
89 \( 1 + (66.0 + 38.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 12.7T + 9.40e3T^{2} \)
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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

−9.394577158397076672880114849845, −8.674869962957973313360462479388, −7.66219574183339445604591859647, −6.51051408517822165288356036902, −6.23917837695123580392231863751, −5.55638719443867523736915926173, −4.23206662661821186014439289437, −3.24957294133283523804437108859, −2.19683193805360667573447413548, −1.42579879241939519933458607247, 0.71003948538679794079584614848, 1.82391658542711681985843711013, 2.57398970070131109525976810496, 4.12777122889700253524017554874, 4.97721355181396992712570722037, 5.55801437315754242632051325877, 6.65933701406235595749529031263, 6.98316138330125536461683309690, 8.614779497925852382142580592356, 9.095276356521278384003575898213

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