↑

Properties

Label 2-5e2-5.4-c25-0-27
Degree $2$
Conductor $25$
Sign $0.447 + 0.894i$
Analytic cond. $98.9991$
Root an. cond. $9.94983$
Motivic weight $25$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 48i·2-s − 1.95e5i·3-s + 3.35e7·4-s + 9.39e6·6-s − 3.90e10i·7-s + 3.22e9i·8-s + 8.08e11·9-s + 8.41e12·11-s − 6.56e12i·12-s − 8.16e13i·13-s + 1.87e12·14-s + 1.12e15·16-s + 2.51e15i·17-s + 3.88e13i·18-s + 6.08e15·19-s + ⋯
L(s)  = 1  + 0.00828i·2-s − 0.212i·3-s + 0.999·4-s + 0.00176·6-s − 1.06i·7-s + 0.0165i·8-s + 0.954·9-s + 0.808·11-s − 0.212i·12-s − 0.972i·13-s + 0.00884·14-s + 0.999·16-s + 1.04i·17-s + 0.00791i·18-s + 0.630·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(98.9991\)
Root analytic conductor: \(9.94983\)
Motivic weight: \(25\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :25/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(13)\) \(\approx\) \(3.634843002\)
\(L(\frac12)\) \(\approx\) \(3.634843002\)
\(L(\frac{27}{2})\) 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 \( 1 - 48iT - 3.35e7T^{2} \)
3 \( 1 + 1.95e5iT - 8.47e11T^{2} \)
7 \( 1 + 3.90e10iT - 1.34e21T^{2} \)
11 \( 1 - 8.41e12T + 1.08e26T^{2} \)
13 \( 1 + 8.16e13iT - 7.05e27T^{2} \)
17 \( 1 - 2.51e15iT - 5.77e30T^{2} \)
19 \( 1 - 6.08e15T + 9.30e31T^{2} \)
23 \( 1 + 9.49e16iT - 1.10e34T^{2} \)
29 \( 1 - 2.71e17T + 3.63e36T^{2} \)
31 \( 1 - 4.29e18T + 1.92e37T^{2} \)
37 \( 1 + 2.03e19iT - 1.60e39T^{2} \)
41 \( 1 + 1.83e20T + 2.08e40T^{2} \)
43 \( 1 - 3.00e20iT - 6.86e40T^{2} \)
47 \( 1 - 9.24e20iT - 6.34e41T^{2} \)
53 \( 1 + 9.90e20iT - 1.27e43T^{2} \)
59 \( 1 + 1.30e22T + 1.86e44T^{2} \)
61 \( 1 - 9.01e21T + 4.29e44T^{2} \)
67 \( 1 - 2.66e22iT - 4.48e45T^{2} \)
71 \( 1 + 1.92e23T + 1.91e46T^{2} \)
73 \( 1 - 4.24e22iT - 3.82e46T^{2} \)
79 \( 1 - 2.71e23T + 2.75e47T^{2} \)
83 \( 1 + 9.31e23iT - 9.48e47T^{2} \)
89 \( 1 - 1.76e24T + 5.42e48T^{2} \)
97 \( 1 + 2.82e24iT - 4.66e49T^{2} \)
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

−12.12008900932182728886820474660, −10.75999455121403711229415765039, −10.00899364632405505313022878538, −8.050155962195873998427181785333, −7.08057749658177703561591762160, −6.16228919840969828852366202382, −4.35203390620115321510580939190, −3.15836819733434756394781710690, −1.60779795351867689517147118310, −0.839110201010052614100169168926, 1.24401322257868636129892952772, 2.20871615100511393397996153632, 3.50987995831921873148194917578, 5.04247148704371041283838971388, 6.42067115473990850310971331411, 7.33676007093481220678349713540, 8.997168779348518884236944855081, 10.03104510430905611961860172574, 11.64666502029399476707814223085, 12.03743822382065575559198500553

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