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Properties

Label 2-42e2-28.27-c1-0-92
Degree $2$
Conductor $1764$
Sign $-0.156 + 0.987i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 2.00·4-s − 4.46i·5-s + 2.82·8-s − 6.30i·10-s − 5.99i·13-s + 4.00·16-s + 4.90i·17-s − 8.92i·20-s − 14.8·25-s − 8.47i·26-s + 4.24·29-s + 5.65·32-s + 6.94i·34-s − 9.89·37-s + ⋯
L(s)  = 1  + 1.00·2-s + 1.00·4-s − 1.99i·5-s + 1.00·8-s − 1.99i·10-s − 1.66i·13-s + 1.00·16-s + 1.19i·17-s − 1.99i·20-s − 2.97·25-s − 1.66i·26-s + 0.787·29-s + 1.00·32-s + 1.19i·34-s − 1.62·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.275323355\)
\(L(\frac12)\) \(\approx\) \(3.275323355\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 4.46iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 5.99iT - 13T^{2} \)
17 \( 1 - 4.90iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 9.89T + 37T^{2} \)
41 \( 1 - 3.56iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 14T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 7.25iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 3.11iT - 89T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−8.853189620922820074602968711524, −8.236426559481734793450533166053, −7.65579649160068106285858282896, −6.34643848136662917521885651798, −5.51530489308676997596878882473, −5.09008853240271252774322057214, −4.20516138410615193357430106547, −3.35773089020623060226404995079, −1.93312794277561664836368702252, −0.867085499316343493622258462187, 1.99151461528811698491132885601, 2.73446442249233607739744540925, 3.58992521362647725679940517189, 4.40581186188239318371383229845, 5.55232767218868948206627730284, 6.43383162578178986285235765968, 7.12417906074245785388418495930, 7.24609529455948873422163951258, 8.681985212085394661508211450045, 9.864899370668239400426830983382

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