↑

Properties

Label 2-52-52.51-c2-0-0
Degree $2$
Conductor $52$
Sign $-0.863 - 0.504i$
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  + (−1.54 + 1.26i)2-s + 5.02i·3-s + (0.780 − 3.92i)4-s + 0.712i·5-s + (−6.37 − 7.76i)6-s − 7.05·7-s + (3.77 + 7.05i)8-s − 16.2·9-s + (−0.903 − 1.10i)10-s + 15.8·11-s + (19.7 + 3.92i)12-s + (−4.24 + 12.2i)13-s + (10.9 − 8.94i)14-s − 3.57·15-s + (−14.7 − 6.12i)16-s + 15.4·17-s + ⋯
L(s)  = 1  + (−0.773 + 0.634i)2-s + 1.67i·3-s + (0.195 − 0.980i)4-s + 0.142i·5-s + (−1.06 − 1.29i)6-s − 1.00·7-s + (0.471 + 0.881i)8-s − 1.80·9-s + (−0.0903 − 0.110i)10-s + 1.44·11-s + (1.64 + 0.326i)12-s + (−0.326 + 0.945i)13-s + (0.778 − 0.639i)14-s − 0.238·15-s + (−0.923 − 0.382i)16-s + 0.911·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $-0.863 - 0.504i$
Analytic conductor: \(1.41689\)
Root analytic conductor: \(1.19033\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1),\ -0.863 - 0.504i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.190051 + 0.701428i\)
\(L(\frac12)\) \(\approx\) \(0.190051 + 0.701428i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.54 - 1.26i)T \)
13 \( 1 + (4.24 - 12.2i)T \)
good3 \( 1 - 5.02iT - 9T^{2} \)
5 \( 1 - 0.712iT - 25T^{2} \)
7 \( 1 + 7.05T + 49T^{2} \)
11 \( 1 - 15.8T + 121T^{2} \)
17 \( 1 - 15.4T + 289T^{2} \)
19 \( 1 - 10.6T + 361T^{2} \)
23 \( 1 - 21.3iT - 529T^{2} \)
29 \( 1 + 2.24T + 841T^{2} \)
31 \( 1 + 26.2T + 961T^{2} \)
37 \( 1 + 42.5iT - 1.36e3T^{2} \)
41 \( 1 + 33.2iT - 1.68e3T^{2} \)
43 \( 1 + 46.4iT - 1.84e3T^{2} \)
47 \( 1 - 47.4T + 2.20e3T^{2} \)
53 \( 1 + 24.7T + 2.80e3T^{2} \)
59 \( 1 - 44.2T + 3.48e3T^{2} \)
61 \( 1 - 91.2T + 3.72e3T^{2} \)
67 \( 1 + 70.7T + 4.48e3T^{2} \)
71 \( 1 + 10.5T + 5.04e3T^{2} \)
73 \( 1 + 52.1iT - 5.32e3T^{2} \)
79 \( 1 + 12.5iT - 6.24e3T^{2} \)
83 \( 1 + 21.4T + 6.88e3T^{2} \)
89 \( 1 - 137. iT - 7.92e3T^{2} \)
97 \( 1 - 137. iT - 9.40e3T^{2} \)
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

−15.99952165443293953601134059328, −14.80679025286838906658768384728, −14.14270799473136460414026176107, −11.79769223505283272247902758702, −10.56287950649291793865346935762, −9.445903636614161969655204131734, −9.127114675499295482239660120413, −6.98468106760241427120409339010, −5.52464490182956834616604364970, −3.78513549682681769156424887071, 1.03070909189775976234699155661, 3.05215249818413951225362625645, 6.37293402475266512079834543152, 7.37121559327858274156281912187, 8.611997583041933216174469303101, 9.897786001555749315214078012908, 11.57275737432252099095185386480, 12.50290158934819025394564161906, 13.02693280516298774926020788857, 14.43384690252951794618366401836

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