Properties

Label 2-42e2-12.11-c1-0-31
Degree $2$
Conductor $1764$
Sign $-0.230 - 0.973i$
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  + (0.581 + 1.28i)2-s + (−1.32 + 1.50i)4-s + (−2.70 − 0.832i)8-s + 6.57·11-s + (−0.5 − 3.96i)16-s + (3.82 + 8.46i)22-s − 1.91·23-s + 5·25-s + 6.06i·29-s + (4.82 − 2.95i)32-s + 10.5·37-s + 12i·43-s + (−8.69 + 9.85i)44-s + (−1.11 − 2.46i)46-s + (2.90 + 6.44i)50-s + ⋯
L(s)  = 1  + (0.411 + 0.911i)2-s + (−0.661 + 0.750i)4-s + (−0.955 − 0.294i)8-s + 1.98·11-s + (−0.125 − 0.992i)16-s + (0.815 + 1.80i)22-s − 0.399·23-s + 25-s + 1.12i·29-s + (0.852 − 0.522i)32-s + 1.73·37-s + 1.82i·43-s + (−1.31 + 1.48i)44-s + (−0.164 − 0.363i)46-s + (0.411 + 0.911i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.230 - 0.973i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.230 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.150981711\)
\(L(\frac12)\) \(\approx\) \(2.150981711\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.581 - 1.28i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 5T^{2} \)
11 \( 1 - 6.57T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 1.91T + 23T^{2} \)
29 \( 1 - 6.06iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 14.5iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 15.8iT - 67T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 15.8iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.269913763569990865207157418429, −8.703356829409532656086977717211, −7.88082215841577220635149648028, −6.86238133918261790600753668418, −6.50179343686056363373409898175, −5.59592070180733887223556195876, −4.55716618244743540308791449847, −3.91994937968984945359937857199, −2.92614850017699024705418321777, −1.20723827344751300427739586131, 0.867907152865859272279176720612, 1.94264850714508135166459931508, 3.11399202297402287835073104776, 4.06539189014804358345370952680, 4.59396540791073106202458533369, 5.88978845345405175028442626771, 6.37775744337147583751793034379, 7.46141682087182213837015012970, 8.673992120882978349621751476954, 9.179123229874955048498817478524

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