Properties

Label 2-42e2-7.4-c3-0-10
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  + (2 + 3.46i)5-s + (−10 + 17.3i)11-s − 4·13-s + (12 − 20.7i)17-s + (−22 − 38.1i)19-s + (36 + 62.3i)23-s + (54.5 − 94.3i)25-s + 38·29-s + (−92 + 159. i)31-s + (15 + 25.9i)37-s + 216·41-s − 164·43-s + (260 + 450. i)47-s + (−73 + 126. i)53-s − 80·55-s + ⋯
L(s)  = 1  + (0.178 + 0.309i)5-s + (−0.274 + 0.474i)11-s − 0.0853·13-s + (0.171 − 0.296i)17-s + (−0.265 − 0.460i)19-s + (0.326 + 0.565i)23-s + (0.435 − 0.755i)25-s + 0.243·29-s + (−0.533 + 0.923i)31-s + (0.0666 + 0.115i)37-s + 0.822·41-s − 0.581·43-s + (0.806 + 1.39i)47-s + (−0.189 + 0.327i)53-s − 0.196·55-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: $\chi_{1764} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.184183480\)
\(L(\frac12)\) \(\approx\) \(1.184183480\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2 - 3.46i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (10 - 17.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 4T + 2.19e3T^{2} \)
17 \( 1 + (-12 + 20.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (22 + 38.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-36 - 62.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 38T + 2.43e4T^{2} \)
31 \( 1 + (92 - 159. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-15 - 25.9i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 216T + 6.89e4T^{2} \)
43 \( 1 + 164T + 7.95e4T^{2} \)
47 \( 1 + (-260 - 450. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (73 - 126. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-230 + 398. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (314 + 543. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (278 - 481. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 592T + 3.57e5T^{2} \)
73 \( 1 + (512 - 886. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-52 - 90.0i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 324T + 5.71e5T^{2} \)
89 \( 1 + (-448 - 775. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 920T + 9.12e5T^{2} \)
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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

−9.258932197644251308061130335448, −8.458925523683063962221149853482, −7.52868337721940108961666650486, −6.90727953252955168851646224053, −6.04874153282798302642696283570, −5.10640391310267851065185162363, −4.34897948332733520431332275877, −3.16420049115166233289312811920, −2.36393009003872537028210473111, −1.14065376453333108388021751581, 0.26253422742641477185379264064, 1.44211578610326664020393202519, 2.56794576055729228163834853399, 3.59921184502487452881769679229, 4.54187640471385713038089645722, 5.48929651679391181385867323296, 6.11662887089841813714241445136, 7.13776326290187642928562331267, 7.899180096897186414646517333749, 8.742950490049630464892116059117

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