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Properties

Label 2-42e2-7.2-c3-0-1
Degree $2$
Conductor $1764$
Sign $-0.605 + 0.795i$
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  + (−7.93 + 13.7i)5-s + (−7.93 − 13.7i)11-s + 26·13-s + (39.6 + 68.7i)17-s + (−34 + 58.8i)19-s + (−23.8 + 41.2i)23-s + (−63.5 − 109. i)25-s − 253.·29-s + (−106 − 183. i)31-s + (−109 + 188. i)37-s + 396.·41-s + 260·43-s + (−206. + 357. i)47-s + (238. + 412. i)53-s + 252·55-s + ⋯
L(s)  = 1  + (−0.709 + 1.22i)5-s + (−0.217 − 0.376i)11-s + 0.554·13-s + (0.566 + 0.980i)17-s + (−0.410 + 0.711i)19-s + (−0.215 + 0.373i)23-s + (−0.508 − 0.879i)25-s − 1.62·29-s + (−0.614 − 1.06i)31-s + (−0.484 + 0.838i)37-s + 1.51·41-s + 0.922·43-s + (−0.640 + 1.10i)47-s + (0.617 + 1.06i)53-s + 0.617·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2187477276\)
\(L(\frac12)\) \(\approx\) \(0.2187477276\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.93 - 13.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (7.93 + 13.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 26T + 2.19e3T^{2} \)
17 \( 1 + (-39.6 - 68.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (34 - 58.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (23.8 - 41.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 253.T + 2.43e4T^{2} \)
31 \( 1 + (106 + 183. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (109 - 188. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 396.T + 6.89e4T^{2} \)
43 \( 1 - 260T + 7.95e4T^{2} \)
47 \( 1 + (206. - 357. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-238. - 412. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-142. - 247. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-161 + 278. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (178 + 308. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.12e3T + 3.57e5T^{2} \)
73 \( 1 + (-113 - 195. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (220 - 381. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 253.T + 5.71e5T^{2} \)
89 \( 1 + (103. - 178. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.33e3T + 9.12e5T^{2} \)
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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

−9.510435576614936291782344521212, −8.501562217958942000813103749307, −7.71730073675868204537974338563, −7.28807492185824103439571408293, −6.07127805331913061246402506489, −5.78833047167547811424385117159, −4.15514592501247466465426756013, −3.64785516421569821366821487357, −2.74358864414720114985497165604, −1.52030078597777683165342806973, 0.05471439942842974204613360136, 0.945436152918830334699302371880, 2.16536164109119294067795391576, 3.49146064558070301311141666565, 4.30900669077086508541689099467, 5.09382990306391556786280602826, 5.78641377561631990616022013055, 7.10830177178368523224722329755, 7.58768074611974818855962747304, 8.630243708809585956291887983248

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