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Properties

Label 2-2184-13.12-c1-0-22
Degree $2$
Conductor $2184$
Sign $0.554 - 0.832i$
Analytic cond. $17.4393$
Root an. cond. $4.17604$
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  + 3-s + 3i·5-s + i·7-s + 9-s − 4i·11-s + (2 − 3i)13-s + 3i·15-s + 6·17-s + 7i·19-s + i·21-s + 23-s − 4·25-s + 27-s + 29-s + 7i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34i·5-s + 0.377i·7-s + 0.333·9-s − 1.20i·11-s + (0.554 − 0.832i)13-s + 0.774i·15-s + 1.45·17-s + 1.60i·19-s + 0.218i·21-s + 0.208·23-s − 0.800·25-s + 0.192·27-s + 0.185·29-s + 1.25i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2184\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.554 - 0.832i$
Analytic conductor: \(17.4393\)
Root analytic conductor: \(4.17604\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2184} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2184,\ (\ :1/2),\ 0.554 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.381227363\)
\(L(\frac12)\) \(\approx\) \(2.381227363\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-2 + 3i)T \)
good5 \( 1 - 3iT - 5T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 7iT - 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 - 7iT - 31T^{2} \)
37 \( 1 + 12iT - 37T^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 6iT - 67T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + 11iT - 73T^{2} \)
79 \( 1 + 15T + 79T^{2} \)
83 \( 1 - 15iT - 83T^{2} \)
89 \( 1 - 11iT - 89T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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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

−9.136268922662755646871534247161, −8.241845737718530276587878812503, −7.78938960437559844348446541166, −6.93598648482174135571576877256, −5.83651899347065251895812312367, −5.66008185649581010367544955315, −3.89716406451601997158302856574, −3.25606840629069950638710258745, −2.71575867438483238539359742483, −1.25398566752504711467160697576, 0.911279507481913780641073198733, 1.85663789636952378524431066620, 3.09652688522753935022968563335, 4.36834244835239648806535266394, 4.59785919233686610937178312113, 5.65575942551248487149151088945, 6.78560093331451575268786882074, 7.50351714796859241958209759049, 8.229153106991404218967781522822, 9.037563669021615013546031226209

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