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Properties

Label 2-42e2-28.27-c1-0-24
Degree $2$
Conductor $1764$
Sign $-0.832 + 0.554i$
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  + (0.856 + 1.12i)2-s + (−0.533 + 1.92i)4-s + 3.85i·5-s + (−2.62 + 1.05i)8-s + (−4.33 + 3.30i)10-s + 1.36i·11-s + 0.369i·13-s + (−3.43 − 2.05i)16-s + 4.50i·17-s − 0.0661·19-s + (−7.43 − 2.05i)20-s + (−1.53 + 1.16i)22-s − 3.20i·23-s − 9.86·25-s + (−0.416 + 0.316i)26-s + ⋯
L(s)  = 1  + (0.605 + 0.795i)2-s + (−0.266 + 0.963i)4-s + 1.72i·5-s + (−0.928 + 0.371i)8-s + (−1.37 + 1.04i)10-s + 0.410i·11-s + 0.102i·13-s + (−0.857 − 0.513i)16-s + 1.09i·17-s − 0.0151·19-s + (−1.66 − 0.459i)20-s + (−0.326 + 0.248i)22-s − 0.669i·23-s − 1.97·25-s + (−0.0816 + 0.0621i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.832 + 0.554i$
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: \((2,\ 1764,\ (\ :1/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.778641409\)
\(L(\frac12)\) \(\approx\) \(1.778641409\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.856 - 1.12i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 3.85iT - 5T^{2} \)
11 \( 1 - 1.36iT - 11T^{2} \)
13 \( 1 - 0.369iT - 13T^{2} \)
17 \( 1 - 4.50iT - 17T^{2} \)
19 \( 1 + 0.0661T + 19T^{2} \)
23 \( 1 + 3.20iT - 23T^{2} \)
29 \( 1 - 3.11T + 29T^{2} \)
31 \( 1 - 6.03T + 31T^{2} \)
37 \( 1 + 5.49T + 37T^{2} \)
41 \( 1 - 8.45iT - 41T^{2} \)
43 \( 1 + 6.30iT - 43T^{2} \)
47 \( 1 + 1.42T + 47T^{2} \)
53 \( 1 - 2.54T + 53T^{2} \)
59 \( 1 - 3.43T + 59T^{2} \)
61 \( 1 - 1.43iT - 61T^{2} \)
67 \( 1 + 9.76iT - 67T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 - 1.80iT - 73T^{2} \)
79 \( 1 - 12.4iT - 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + 1.29iT - 89T^{2} \)
97 \( 1 - 2.88iT - 97T^{2} \)
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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

−9.899128826420824569367925602911, −8.697136324613266759072453066925, −7.965073317314221941908949265598, −7.17544043659906945791164372131, −6.50996395734840267516783155306, −6.08866351625670431005760010204, −4.88470238832769012077041508809, −3.92351726838100883168858161247, −3.12670426843768229279322485222, −2.23241310953148951405177037529, 0.54722814817202318657917996713, 1.47937107624136169135333741134, 2.75376166342849971302876794000, 3.87614440003906507050441646544, 4.72648565984183770307927711168, 5.27728513767267424014116318330, 6.02378244412134971233371732817, 7.22706994491196349218072895095, 8.424808980340091369354163962197, 8.878091852210881481899078764391

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