Properties

Label 2-6384-1.1-c1-0-36
Degree $2$
Conductor $6384$
Sign $1$
Analytic cond. $50.9764$
Root an. cond. $7.13978$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 1.69·5-s + 7-s + 9-s + 1.32·11-s − 1.69·15-s + 6.14·17-s + 19-s + 21-s − 2.91·23-s − 2.12·25-s + 27-s − 1.42·29-s + 3.32·31-s + 1.32·33-s − 1.69·35-s + 9.83·37-s − 8.04·41-s − 7.16·43-s − 1.69·45-s + 7.66·47-s + 49-s + 6.14·51-s + 8.81·53-s − 2.24·55-s + 57-s − 9.43·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.758·5-s + 0.377·7-s + 0.333·9-s + 0.399·11-s − 0.437·15-s + 1.49·17-s + 0.229·19-s + 0.218·21-s − 0.608·23-s − 0.424·25-s + 0.192·27-s − 0.264·29-s + 0.597·31-s + 0.230·33-s − 0.286·35-s + 1.61·37-s − 1.25·41-s − 1.09·43-s − 0.252·45-s + 1.11·47-s + 0.142·49-s + 0.860·51-s + 1.21·53-s − 0.302·55-s + 0.132·57-s − 1.22·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6384\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(50.9764\)
Root analytic conductor: \(7.13978\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6384} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6384,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.424550942\)
\(L(\frac12)\) \(\approx\) \(2.424550942\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - T \)
19 \( 1 - T \)
good5 \( 1 + 1.69T + 5T^{2} \)
11 \( 1 - 1.32T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 6.14T + 17T^{2} \)
23 \( 1 + 2.91T + 23T^{2} \)
29 \( 1 + 1.42T + 29T^{2} \)
31 \( 1 - 3.32T + 31T^{2} \)
37 \( 1 - 9.83T + 37T^{2} \)
41 \( 1 + 8.04T + 41T^{2} \)
43 \( 1 + 7.16T + 43T^{2} \)
47 \( 1 - 7.66T + 47T^{2} \)
53 \( 1 - 8.81T + 53T^{2} \)
59 \( 1 + 9.43T + 59T^{2} \)
61 \( 1 - 4.64T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 - 3.22T + 71T^{2} \)
73 \( 1 + 8.04T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 4.77T + 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 6.71T + 97T^{2} \)
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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

−8.004552010067865867022339405973, −7.55872928431029365576081702940, −6.77279425163237071833358698480, −5.87879314160598569892058612597, −5.12087708285099732239618560025, −4.18859739120249070734800571675, −3.67967074092403567413569126986, −2.88082640076578093533578466539, −1.82864275415356968922651414554, −0.811402725924341414403598785800, 0.811402725924341414403598785800, 1.82864275415356968922651414554, 2.88082640076578093533578466539, 3.67967074092403567413569126986, 4.18859739120249070734800571675, 5.12087708285099732239618560025, 5.87879314160598569892058612597, 6.77279425163237071833358698480, 7.55872928431029365576081702940, 8.004552010067865867022339405973

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