↑

Properties

Label 2-5e2-5.2-c32-0-14
Degree $2$
Conductor $25$
Sign $0.850 + 0.525i$
Analytic cond. $162.166$
Root an. cond. $12.7344$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−6.89e4 − 6.89e4i)2-s + (−5.43e7 + 5.43e7i)3-s + 5.20e9i·4-s + 7.48e12·6-s + (−3.40e13 − 3.40e13i)7-s + (6.30e13 − 6.30e13i)8-s − 4.04e15i·9-s − 2.53e16·11-s + (−2.82e17 − 2.82e17i)12-s + (3.66e17 − 3.66e17i)13-s + 4.69e18i·14-s + 1.36e19·16-s + (5.02e19 + 5.02e19i)17-s + (−2.78e20 + 2.78e20i)18-s + 6.86e19i·19-s + ⋯
L(s)  = 1  + (−1.05 − 1.05i)2-s + (−1.26 + 1.26i)3-s + 1.21i·4-s + 2.65·6-s + (−1.02 − 1.02i)7-s + (0.224 − 0.224i)8-s − 2.18i·9-s − 0.551·11-s + (−1.53 − 1.53i)12-s + (0.551 − 0.551i)13-s + 2.15i·14-s + 0.741·16-s + (1.03 + 1.03i)17-s + (−2.29 + 2.29i)18-s + 0.238i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(162.166\)
Root analytic conductor: \(12.7344\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :16),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.3562496920\)
\(L(\frac12)\) \(\approx\) \(0.3562496920\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (6.89e4 + 6.89e4i)T + 4.29e9iT^{2} \)
3 \( 1 + (5.43e7 - 5.43e7i)T - 1.85e15iT^{2} \)
7 \( 1 + (3.40e13 + 3.40e13i)T + 1.10e27iT^{2} \)
11 \( 1 + 2.53e16T + 2.11e33T^{2} \)
13 \( 1 + (-3.66e17 + 3.66e17i)T - 4.42e35iT^{2} \)
17 \( 1 + (-5.02e19 - 5.02e19i)T + 2.36e39iT^{2} \)
19 \( 1 - 6.86e19iT - 8.31e40T^{2} \)
23 \( 1 + (-1.56e21 + 1.56e21i)T - 3.76e43iT^{2} \)
29 \( 1 - 6.33e22iT - 6.26e46T^{2} \)
31 \( 1 - 5.37e23T + 5.29e47T^{2} \)
37 \( 1 + (-4.80e24 - 4.80e24i)T + 1.52e50iT^{2} \)
41 \( 1 + 7.84e25T + 4.06e51T^{2} \)
43 \( 1 + (6.30e25 - 6.30e25i)T - 1.86e52iT^{2} \)
47 \( 1 + (2.58e26 + 2.58e26i)T + 3.21e53iT^{2} \)
53 \( 1 + (-4.72e27 + 4.72e27i)T - 1.50e55iT^{2} \)
59 \( 1 + 2.35e28iT - 4.64e56T^{2} \)
61 \( 1 + 2.89e28T + 1.35e57T^{2} \)
67 \( 1 + (-6.00e28 - 6.00e28i)T + 2.71e58iT^{2} \)
71 \( 1 + 3.68e28T + 1.73e59T^{2} \)
73 \( 1 + (1.90e29 - 1.90e29i)T - 4.22e59iT^{2} \)
79 \( 1 + 3.04e30iT - 5.29e60T^{2} \)
83 \( 1 + (1.94e30 - 1.94e30i)T - 2.57e61iT^{2} \)
89 \( 1 - 2.88e31iT - 2.40e62T^{2} \)
97 \( 1 + (-1.59e31 - 1.59e31i)T + 3.77e63iT^{2} \)
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

−10.77758248037641751535753510554, −10.24501296364715280724900186204, −9.744756922233759465443781260996, −8.243626692616512617989557069090, −6.45739396255652328425079343204, −5.32928828222916479654376284875, −3.83193993193442932868525126751, −3.18079243556311701921433300130, −1.19397818732872377045984245331, −0.38171302878250293652391030506, 0.36477850648909819852223620335, 1.31759473527116864155222756464, 2.79778712180814490588230329488, 5.33306380982969625939558863264, 6.04004610848245237849720433153, 6.82642884029858529243174138969, 7.68897521331591696818830924858, 8.919437517405525708855431957874, 10.11478370641045396530881996906, 11.65289505939566067548317352370

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