↑

Properties

Label 2-1216-152.37-c2-0-74
Degree $2$
Conductor $1216$
Sign $-0.549 + 0.835i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.88·3-s − 4.10i·5-s + 9.53·7-s − 0.676·9-s − 13.4i·11-s − 24.6·13-s − 11.8i·15-s − 3.18·17-s + (5.97 − 18.0i)19-s + 27.5·21-s − 13.9·23-s + 8.16·25-s − 27.9·27-s − 37.6·29-s − 26.8i·31-s + ⋯
L(s)  = 1  + 0.961·3-s − 0.820i·5-s + 1.36·7-s − 0.0752·9-s − 1.22i·11-s − 1.89·13-s − 0.789i·15-s − 0.187·17-s + (0.314 − 0.949i)19-s + 1.31·21-s − 0.607·23-s + 0.326·25-s − 1.03·27-s − 1.29·29-s − 0.865i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.549 + 0.835i$
Analytic conductor: \(33.1336\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1),\ -0.549 + 0.835i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.099296679\)
\(L(\frac12)\) \(\approx\) \(2.099296679\)
\(L(2)\) 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 \( 1 + (-5.97 + 18.0i)T \)
good3 \( 1 - 2.88T + 9T^{2} \)
5 \( 1 + 4.10iT - 25T^{2} \)
7 \( 1 - 9.53T + 49T^{2} \)
11 \( 1 + 13.4iT - 121T^{2} \)
13 \( 1 + 24.6T + 169T^{2} \)
17 \( 1 + 3.18T + 289T^{2} \)
23 \( 1 + 13.9T + 529T^{2} \)
29 \( 1 + 37.6T + 841T^{2} \)
31 \( 1 + 26.8iT - 961T^{2} \)
37 \( 1 - 7.92T + 1.36e3T^{2} \)
41 \( 1 + 24.3iT - 1.68e3T^{2} \)
43 \( 1 - 44.2iT - 1.84e3T^{2} \)
47 \( 1 - 2.67T + 2.20e3T^{2} \)
53 \( 1 + 58.2T + 2.80e3T^{2} \)
59 \( 1 - 99.5T + 3.48e3T^{2} \)
61 \( 1 + 55.1iT - 3.72e3T^{2} \)
67 \( 1 - 60.1T + 4.48e3T^{2} \)
71 \( 1 - 65.8iT - 5.04e3T^{2} \)
73 \( 1 - 115.T + 5.32e3T^{2} \)
79 \( 1 - 89.0iT - 6.24e3T^{2} \)
83 \( 1 + 29.1iT - 6.88e3T^{2} \)
89 \( 1 - 147. iT - 7.92e3T^{2} \)
97 \( 1 + 170. iT - 9.40e3T^{2} \)
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

−9.224454116102703135338509563125, −8.241154160481885814382364529668, −8.044480059631700376533909083501, −7.06708399101685048138153147513, −5.56696871026268817516684038804, −5.00551827402104864016144525363, −4.06119451108949654702344583762, −2.78968258702391706544005371245, −1.95844863325738697400087036138, −0.48992644117846723339652626719, 1.90688759347001227633041196209, 2.38725569038710141164835121504, 3.58021637355357804769276641799, 4.67042752540583757376473387221, 5.40326981430538605847704421461, 6.84038568366017595840862143380, 7.61165237883639468550163649366, 7.924893118811735536148875389165, 9.021580755724465844071860549017, 9.810392976678826771015992563063

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