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Properties

Label 2-42e2-7.4-c3-0-0
Degree $2$
Conductor $1764$
Sign $-0.386 + 0.922i$
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  + (6.41 + 11.1i)5-s + (−18.4 + 31.8i)11-s − 87.1·13-s + (−51.2 + 88.8i)17-s + (47.9 + 82.9i)19-s + (−48 − 83.1i)23-s + (−19.7 + 34.1i)25-s + 212.·29-s + (−79.6 + 137. i)31-s + (−64.3 − 111. i)37-s − 298.·41-s − 33.3·43-s + (−135. − 234. i)47-s + (224. − 388. i)53-s − 472.·55-s + ⋯
L(s)  = 1  + (0.573 + 0.993i)5-s + (−0.504 + 0.874i)11-s − 1.85·13-s + (−0.731 + 1.26i)17-s + (0.578 + 1.00i)19-s + (−0.435 − 0.753i)23-s + (−0.157 + 0.273i)25-s + 1.35·29-s + (−0.461 + 0.799i)31-s + (−0.285 − 0.495i)37-s − 1.13·41-s − 0.118·43-s + (−0.420 − 0.728i)47-s + (0.580 − 1.00i)53-s − 1.15·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.07004986676\)
\(L(\frac12)\) \(\approx\) \(0.07004986676\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.41 - 11.1i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (18.4 - 31.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 87.1T + 2.19e3T^{2} \)
17 \( 1 + (51.2 - 88.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-47.9 - 82.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (48 + 83.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 212.T + 2.43e4T^{2} \)
31 \( 1 + (79.6 - 137. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (64.3 + 111. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 298.T + 6.89e4T^{2} \)
43 \( 1 + 33.3T + 7.95e4T^{2} \)
47 \( 1 + (135. + 234. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-224. + 388. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-334. + 578. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (121. + 211. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-167. + 290. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 339.T + 3.57e5T^{2} \)
73 \( 1 + (459. - 795. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-68.1 - 118. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 287.T + 5.71e5T^{2} \)
89 \( 1 + (80.9 + 140. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 182.T + 9.12e5T^{2} \)
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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

−9.948762224367873208027991062381, −8.613203565521010074849041956638, −7.84445690669156393474188103588, −6.91690905599551012145017189151, −6.54304935545633662765765472653, −5.38487488195630563086593122026, −4.67657053184224337980990235154, −3.51746667718445164263691494161, −2.41710700516834648384205009792, −1.90729115789214363316616785034, 0.01573424633587009241594944442, 0.946298588055574099817432559619, 2.31133330988529470642714989423, 3.02536360208678009966167139979, 4.60702257484864962288527794436, 5.02650845861444621045935303205, 5.74789911969188066111380142984, 6.92211136906902551639488293991, 7.56712791661165500667682504756, 8.534762458657847439319656724681

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