Properties

Label 2-1216-152.37-c0-0-4
Degree $2$
Conductor $1216$
Sign $0.258 + 0.965i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·5-s + 1.73·7-s − 9-s i·11-s − 17-s + i·19-s − 1.99·25-s − 2.99i·35-s + i·43-s + 1.73i·45-s + 1.73·47-s + 1.99·49-s − 1.73·55-s − 1.73i·61-s − 1.73·63-s + ⋯
L(s)  = 1  − 1.73i·5-s + 1.73·7-s − 9-s i·11-s − 17-s + i·19-s − 1.99·25-s − 2.99i·35-s + i·43-s + 1.73i·45-s + 1.73·47-s + 1.99·49-s − 1.73·55-s − 1.73i·61-s − 1.73·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.258 + 0.965i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :0),\ 0.258 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.125899593\)
\(L(\frac12)\) \(\approx\) \(1.125899593\)
\(L(1)\) 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 \( 1 - iT \)
good3 \( 1 + T^{2} \)
5 \( 1 + 1.73iT - T^{2} \)
7 \( 1 - 1.73T + T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 - 1.73T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 1.73iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 2iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.468393245248311978610028697593, −8.644790175219062476372765591033, −8.365483112523258028602134831817, −7.73874528907677206285809350335, −6.10944500377023084003472926039, −5.37173359460988602407981346933, −4.77758906319748045592440684855, −3.86138599817062274226591239329, −2.18897370204960924819179293151, −1.06166890326627544015399119234, 2.08601991965922016682182788152, 2.68253133036454636010506112612, 4.05939879201996381022235652347, 4.97994506259223041681830704473, 5.96936623512845364835318545403, 7.02422408917260611142478651111, 7.44241430682648214293199617657, 8.429447277324166384827487973601, 9.204714783875373542955193355639, 10.48698786022753658915608073535

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