↑

Properties

Label 2-52e2-52.3-c0-0-2
Degree $2$
Conductor $2704$
Sign $0.979 - 0.202i$
Analytic cond. $1.34947$
Root an. cond. $1.16166$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·5-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)17-s + 1.99·25-s + (−0.5 − 0.866i)29-s + (0.866 + 1.5i)37-s + (0.866 + 1.5i)41-s + (−0.866 + 1.49i)45-s + (−0.5 − 0.866i)49-s − 53-s + (0.5 − 0.866i)61-s − 1.73·73-s + (−0.499 − 0.866i)81-s + (0.866 − 1.49i)85-s + (−0.5 − 0.866i)101-s + ⋯
L(s)  = 1  + 1.73·5-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)17-s + 1.99·25-s + (−0.5 − 0.866i)29-s + (0.866 + 1.5i)37-s + (0.866 + 1.5i)41-s + (−0.866 + 1.49i)45-s + (−0.5 − 0.866i)49-s − 53-s + (0.5 − 0.866i)61-s − 1.73·73-s + (−0.499 − 0.866i)81-s + (0.866 − 1.49i)85-s + (−0.5 − 0.866i)101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2704\)    =    \(2^{4} \cdot 13^{2}\)
Sign: $0.979 - 0.202i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2704} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2704,\ (\ :0),\ 0.979 - 0.202i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.638844559\)
\(L(\frac12)\) \(\approx\) \(1.638844559\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 - 1.73T + T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + 1.73T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−9.237641971838884093740600672749, −8.295598311534669541961300562774, −7.60812109621593937038886998318, −6.53673115864031877775453167931, −5.96114768619780047726786095013, −5.23465828941348454194816414249, −4.60840142593445696350060550709, −3.04538446410200806329768578806, −2.40139316501046747845449482333, −1.41543856381546898403129123601, 1.27722070311046767448861502659, 2.26456635192596139768518558749, 3.19662328718842042164894759121, 4.22993959354043354781116318043, 5.53568200006600218775617410011, 5.77518203401610866432729949979, 6.51743887968901961346963159700, 7.37275729676586085398993151656, 8.445686866480236004413682216354, 9.276550614043818153025767134962

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