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Properties

Label 2-42e2-21.17-c3-0-20
Degree $2$
Conductor $1764$
Sign $0.154 - 0.987i$
Analytic cond. $104.079$
Root an. cond. $10.2019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.41 + 5.91i)5-s + (50.5 + 29.1i)11-s + 38.5i·13-s + (16.1 − 27.9i)17-s + (−107. + 62.2i)19-s + (174. − 100. i)23-s + (39.2 − 67.9i)25-s + 104. i·29-s + (240. + 138. i)31-s + (23.8 + 41.2i)37-s − 387.·41-s + 272.·43-s + (−81.5 − 141. i)47-s + (313. + 181. i)53-s + 398. i·55-s + ⋯
L(s)  = 1  + (0.305 + 0.528i)5-s + (1.38 + 0.799i)11-s + 0.822i·13-s + (0.230 − 0.398i)17-s + (−1.30 + 0.751i)19-s + (1.57 − 0.911i)23-s + (0.313 − 0.543i)25-s + 0.668i·29-s + (1.39 + 0.805i)31-s + (0.105 + 0.183i)37-s − 1.47·41-s + 0.966·43-s + (−0.253 − 0.438i)47-s + (0.813 + 0.469i)53-s + 0.976i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.154 - 0.987i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ 0.154 - 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.548429845\)
\(L(\frac12)\) \(\approx\) \(2.548429845\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.41 - 5.91i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-50.5 - 29.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 38.5iT - 2.19e3T^{2} \)
17 \( 1 + (-16.1 + 27.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (107. - 62.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-174. + 100. i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 104. iT - 2.43e4T^{2} \)
31 \( 1 + (-240. - 138. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-23.8 - 41.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 387.T + 6.89e4T^{2} \)
43 \( 1 - 272.T + 7.95e4T^{2} \)
47 \( 1 + (81.5 + 141. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-313. - 181. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (105. - 183. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-202. + 117. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (262. - 454. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 348. iT - 3.57e5T^{2} \)
73 \( 1 + (465. + 268. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (362. + 628. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 392.T + 5.71e5T^{2} \)
89 \( 1 + (-430. - 744. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 978. iT - 9.12e5T^{2} \)
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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

−8.908055663130033379123872262680, −8.662655069042611475058939812980, −7.25159464606937446364675777581, −6.69069296380100476896898967671, −6.22386162517426120391673872737, −4.83025420536942906525684856510, −4.25361024568307260608416525718, −3.12565849458020770469680984635, −2.09722260123143320094857092397, −1.11208551543194347190842860167, 0.61586191392768813900852756434, 1.41918850168401296130606183826, 2.76726448894536805544216595806, 3.72595052591563581864520499777, 4.64861357600531923075573374066, 5.57541397810351206108957002126, 6.30537721853145590369840894155, 7.07676976490283348035711726991, 8.183191405829681101564153821652, 8.795434489858962835854693166423

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