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Properties

Label 2-42e2-12.11-c1-0-32
Degree $2$
Conductor $1764$
Sign $0.944 - 0.328i$
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  + (1.13 − 0.850i)2-s + (0.554 − 1.92i)4-s + 3.87i·5-s + (−1.00 − 2.64i)8-s + (3.29 + 4.37i)10-s + 3.46·11-s − 0.296·13-s + (−3.38 − 2.12i)16-s − 1.56i·17-s + 7.07i·19-s + (7.43 + 2.14i)20-s + (3.92 − 2.95i)22-s + 5.43·23-s − 9.98·25-s + (−0.335 + 0.252i)26-s + ⋯
L(s)  = 1  + (0.799 − 0.601i)2-s + (0.277 − 0.960i)4-s + 1.73i·5-s + (−0.356 − 0.934i)8-s + (1.04 + 1.38i)10-s + 1.04·11-s − 0.0822·13-s + (−0.846 − 0.532i)16-s − 0.380i·17-s + 1.62i·19-s + (1.66 + 0.479i)20-s + (0.835 − 0.628i)22-s + 1.13·23-s − 1.99·25-s + (−0.0657 + 0.0494i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.944 - 0.328i$
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: \((2,\ 1764,\ (\ :1/2),\ 0.944 - 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.796367421\)
\(L(\frac12)\) \(\approx\) \(2.796367421\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 + 0.850i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 3.87iT - 5T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + 0.296T + 13T^{2} \)
17 \( 1 + 1.56iT - 17T^{2} \)
19 \( 1 - 7.07iT - 19T^{2} \)
23 \( 1 - 5.43T + 23T^{2} \)
29 \( 1 - 6.85iT - 29T^{2} \)
31 \( 1 - 2.81iT - 31T^{2} \)
37 \( 1 - 2.51T + 37T^{2} \)
41 \( 1 + 3.55iT - 41T^{2} \)
43 \( 1 - 0.682iT - 43T^{2} \)
47 \( 1 - 2.36T + 47T^{2} \)
53 \( 1 - 0.623iT - 53T^{2} \)
59 \( 1 - 8.85T + 59T^{2} \)
61 \( 1 - 2.66T + 61T^{2} \)
67 \( 1 - 10.6iT - 67T^{2} \)
71 \( 1 - 0.539T + 71T^{2} \)
73 \( 1 - 7.39T + 73T^{2} \)
79 \( 1 - 6.16iT - 79T^{2} \)
83 \( 1 + 6.15T + 83T^{2} \)
89 \( 1 + 11.7iT - 89T^{2} \)
97 \( 1 + 6.84T + 97T^{2} \)
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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

−9.705240698998327678200933296764, −8.708441640238495575940497878513, −7.31928281059214022776203974388, −6.86328748493082309451906837053, −6.13461091517132669666929251841, −5.29945313663600278338069248554, −4.01639638758832287317005297242, −3.40805653187115198719691723167, −2.60704263272022096351847543546, −1.44687384864285138411337412489, 0.868071344142933998047553800645, 2.30928439750821853264113592489, 3.70359110056554558672989112692, 4.52469738161022477999114921402, 5.00435448859252307812110923703, 5.93853195961888175228123222467, 6.72872510598237770579744279288, 7.66002203060157547381323231581, 8.496515178212376979036084666645, 9.036280854162042179919519276334

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