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Properties

Label 2-42e2-12.11-c1-0-29
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

\(L(1)\) \(\approx\) \(2.748534134\)
\(L(\frac12)\) \(\approx\) \(2.748534134\)
\(L(\frac{3}{2})\) 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

−9.686670787221588469281236758404, −8.699645659877113799073922548717, −7.64577176138257002832489361986, −7.21477307808281976100139195372, −6.25466219466369123139420653067, −6.04194357830196414112217554301, −4.68267039693853476524151453258, −3.65479010069798224841698077255, −3.15632596241873434897149308152, −2.01253190379537618604101417863, 0.820981595397107216075105430025, 1.60464812140013274917322500917, 2.99379285124073264670531118354, 4.13219632552512990587634400234, 4.72795623061400133019786268829, 5.37050197868599649741125235890, 6.33678356212784586078048556706, 7.17992821739738742144927910901, 8.463926352619188624573617412111, 9.189845756843400772733715141055

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