Properties

Label 2-1216-19.18-c2-0-63
Degree $2$
Conductor $1216$
Sign $0.0313 + 0.999i$
Analytic cond. $33.1336$
Root an. cond. $5.75617$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.27i·3-s + 7.92·5-s + 5.75·7-s − 9.24·9-s + 16.5·11-s − 7.95i·13-s − 33.8i·15-s − 18.1·17-s + (0.595 + 18.9i)19-s − 24.5i·21-s + 7.02·23-s + 37.7·25-s + 1.04i·27-s − 30.4i·29-s + 34.9i·31-s + ⋯
L(s)  = 1  − 1.42i·3-s + 1.58·5-s + 0.822·7-s − 1.02·9-s + 1.50·11-s − 0.612i·13-s − 2.25i·15-s − 1.06·17-s + (0.0313 + 0.999i)19-s − 1.17i·21-s + 0.305·23-s + 1.50·25-s + 0.0387i·27-s − 1.05i·29-s + 1.12i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0313 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0313 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.0313 + 0.999i$
Analytic conductor: \(33.1336\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1),\ 0.0313 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.233562429\)
\(L(\frac12)\) \(\approx\) \(3.233562429\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-0.595 - 18.9i)T \)
good3 \( 1 + 4.27iT - 9T^{2} \)
5 \( 1 - 7.92T + 25T^{2} \)
7 \( 1 - 5.75T + 49T^{2} \)
11 \( 1 - 16.5T + 121T^{2} \)
13 \( 1 + 7.95iT - 169T^{2} \)
17 \( 1 + 18.1T + 289T^{2} \)
23 \( 1 - 7.02T + 529T^{2} \)
29 \( 1 + 30.4iT - 841T^{2} \)
31 \( 1 - 34.9iT - 961T^{2} \)
37 \( 1 + 58.6iT - 1.36e3T^{2} \)
41 \( 1 - 40.7iT - 1.68e3T^{2} \)
43 \( 1 + 12.5T + 1.84e3T^{2} \)
47 \( 1 - 88.8T + 2.20e3T^{2} \)
53 \( 1 - 62.8iT - 2.80e3T^{2} \)
59 \( 1 + 51.2iT - 3.48e3T^{2} \)
61 \( 1 - 68.1T + 3.72e3T^{2} \)
67 \( 1 - 82.5iT - 4.48e3T^{2} \)
71 \( 1 - 21.9iT - 5.04e3T^{2} \)
73 \( 1 + 57.3T + 5.32e3T^{2} \)
79 \( 1 + 88.3iT - 6.24e3T^{2} \)
83 \( 1 - 0.720T + 6.88e3T^{2} \)
89 \( 1 + 30.1iT - 7.92e3T^{2} \)
97 \( 1 - 169. iT - 9.40e3T^{2} \)
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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

−9.182244716202878699152621782125, −8.549091598975789419842991984704, −7.58780526743342855548431647701, −6.72918019345242431524749333887, −6.14081120378019798075659942890, −5.45038367939128542496712639632, −4.16344113634485882388006499596, −2.53368760194414098337303679142, −1.73001906677043213735925019252, −1.09088548609037308245604612564, 1.42327820629775891484352913849, 2.45127091884934280105056731604, 3.85082099007628164929697961481, 4.65914000604621253765255495382, 5.27917673533278026685722418522, 6.33013623920618025346495397252, 6.99728056188029053423974945213, 8.743972328461705523992824062425, 9.029491474359277581206378311414, 9.650207470967527059191006791631

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