↑

Properties

Label 2-52-13.5-c2-0-1
Degree $2$
Conductor $52$
Sign $0.973 + 0.230i$
Analytic cond. $1.41689$
Root an. cond. $1.19033$
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  − 0.286·3-s + (5.81 − 5.81i)5-s + (8.10 + 8.10i)7-s − 8.91·9-s + (−3.57 − 3.57i)11-s + (−11.2 − 6.52i)13-s + (−1.66 + 1.66i)15-s + 13.0i·17-s + (−12.7 + 12.7i)19-s + (−2.32 − 2.32i)21-s + 6.36i·23-s − 42.6i·25-s + 5.13·27-s + 21.9·29-s + (−29.6 + 29.6i)31-s + ⋯
L(s)  = 1  − 0.0956·3-s + (1.16 − 1.16i)5-s + (1.15 + 1.15i)7-s − 0.990·9-s + (−0.324 − 0.324i)11-s + (−0.864 − 0.502i)13-s + (−0.111 + 0.111i)15-s + 0.768i·17-s + (−0.669 + 0.669i)19-s + (−0.110 − 0.110i)21-s + 0.276i·23-s − 1.70i·25-s + 0.190·27-s + 0.755·29-s + (−0.957 + 0.957i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(1.41689\)
Root analytic conductor: \(1.19033\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1),\ 0.973 + 0.230i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.26362 - 0.147331i\)
\(L(\frac12)\) \(\approx\) \(1.26362 - 0.147331i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (11.2 + 6.52i)T \)
good3 \( 1 + 0.286T + 9T^{2} \)
5 \( 1 + (-5.81 + 5.81i)T - 25iT^{2} \)
7 \( 1 + (-8.10 - 8.10i)T + 49iT^{2} \)
11 \( 1 + (3.57 + 3.57i)T + 121iT^{2} \)
17 \( 1 - 13.0iT - 289T^{2} \)
19 \( 1 + (12.7 - 12.7i)T - 361iT^{2} \)
23 \( 1 - 6.36iT - 529T^{2} \)
29 \( 1 - 21.9T + 841T^{2} \)
31 \( 1 + (29.6 - 29.6i)T - 961iT^{2} \)
37 \( 1 + (2.18 + 2.18i)T + 1.36e3iT^{2} \)
41 \( 1 + (-37.4 + 37.4i)T - 1.68e3iT^{2} \)
43 \( 1 + 40.8iT - 1.84e3T^{2} \)
47 \( 1 + (55.0 + 55.0i)T + 2.20e3iT^{2} \)
53 \( 1 - 25.5T + 2.80e3T^{2} \)
59 \( 1 + (-43.7 - 43.7i)T + 3.48e3iT^{2} \)
61 \( 1 - 9.03T + 3.72e3T^{2} \)
67 \( 1 + (2.92 - 2.92i)T - 4.48e3iT^{2} \)
71 \( 1 + (40.3 - 40.3i)T - 5.04e3iT^{2} \)
73 \( 1 + (-5.90 - 5.90i)T + 5.32e3iT^{2} \)
79 \( 1 - 61.5T + 6.24e3T^{2} \)
83 \( 1 + (-65.8 + 65.8i)T - 6.88e3iT^{2} \)
89 \( 1 + (-41.0 - 41.0i)T + 7.92e3iT^{2} \)
97 \( 1 + (20.4 - 20.4i)T - 9.40e3iT^{2} \)
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

−14.99330983416783653191757467798, −14.12886672192236428188595617781, −12.77523672038533772867937459181, −11.93559344638638146048409851071, −10.44703974007478377639427387907, −8.912968814114390852801317787842, −8.303647766657211810158833853750, −5.73971649256929074498704884626, −5.17432520725426731136819899914, −2.09698581868001836849497720554, 2.45364013604948083207563392452, 4.86786713577332486197467077037, 6.51763681655266599213693436884, 7.70218443848781869087420080236, 9.535511954536058365117633780385, 10.70125626845996680324625154696, 11.41532536743415688068716562894, 13.33620285468402492044357034200, 14.39545399080884393433502916043, 14.63302730919020330139818369424

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