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Properties

Label 2-78e2-13.12-c1-0-11
Degree $2$
Conductor $6084$
Sign $-0.832 - 0.554i$
Analytic cond. $48.5809$
Root an. cond. $6.97000$
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  + 4i·7-s + 8i·19-s + 5·25-s − 4i·31-s + 10i·37-s − 8·43-s − 9·49-s + 14·61-s − 16i·67-s + 10i·73-s − 4·79-s + 14i·97-s − 20·103-s + 2i·109-s + ⋯
L(s)  = 1  + 1.51i·7-s + 1.83i·19-s + 25-s − 0.718i·31-s + 1.64i·37-s − 1.21·43-s − 1.28·49-s + 1.79·61-s − 1.95i·67-s + 1.17i·73-s − 0.450·79-s + 1.42i·97-s − 1.97·103-s + 0.191i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6084\)    =    \(2^{2} \cdot 3^{2} \cdot 13^{2}\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(48.5809\)
Root analytic conductor: \(6.97000\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6084} (4393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6084,\ (\ :1/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.347172990\)
\(L(\frac12)\) \(\approx\) \(1.347172990\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 16iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−8.293633623138981096083602051374, −7.923910457192136484061136063416, −6.76247526882677110359979041715, −6.22152767011717942335812086561, −5.47744102515072723492551368405, −4.96007240891790709671959869047, −3.88411338878443797777786163483, −3.06926239365644243646826894672, −2.25708440520684137637353901768, −1.36929780795719399186590883105, 0.35977859777929516042397977498, 1.22981292813042603918188611955, 2.48111773564975026760910481187, 3.36181574165636352213402989234, 4.16334970435739794849098775774, 4.79741745843610347931844982540, 5.53051310890897367445502908223, 6.81197010536057948825957930213, 6.88599789036389883834982008564, 7.62530348461987855524789813364

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