Properties

Label 2-72e2-8.5-c1-0-60
Degree $2$
Conductor $5184$
Sign $0.707 + 0.707i$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
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  − 0.550i·11-s + 7.89·17-s − 6.34i·19-s + 5·25-s − 6.79·41-s + 12.3i·43-s − 7·49-s − 15.2i·59-s + 0.348i·67-s − 15.6·73-s − 18i·83-s + 18·89-s + 9.69·97-s + 14.1i·107-s + 18·113-s + ⋯
L(s)  = 1  − 0.165i·11-s + 1.91·17-s − 1.45i·19-s + 25-s − 1.06·41-s + 1.88i·43-s − 49-s − 1.98i·59-s + 0.0425i·67-s − 1.83·73-s − 1.97i·83-s + 1.90·89-s + 0.984·97-s + 1.36i·107-s + 1.69·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (2593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.976593921\)
\(L(\frac12)\) \(\approx\) \(1.976593921\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 0.550iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 7.89T + 17T^{2} \)
19 \( 1 + 6.34iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 6.79T + 41T^{2} \)
43 \( 1 - 12.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 15.2iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 0.348iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 18iT - 83T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 - 9.69T + 97T^{2} \)
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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

−8.063798025436533954337913475067, −7.46776100489292448906640457552, −6.67754799227442316497968354130, −6.00796228886776588811722399812, −5.07702938313688218143624007847, −4.63820419933221841008478642101, −3.34337803067026518924717734309, −2.98172640407807164500949864115, −1.67343209789532726374846175922, −0.63837602547448816193466052473, 0.997806460989646868888582719766, 1.91629668148390267814323180266, 3.13965058866266768045650602992, 3.66085489897178372210763034627, 4.67435000513673028117727693744, 5.51212888755915875708811804978, 5.98068714180352262520574206623, 7.00716824502958814139309809091, 7.55807688866920765192140426851, 8.298980743647979808943865653865

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