↑

Properties

Label 2-5e2-5.4-c33-0-25
Degree $2$
Conductor $25$
Sign $-0.894 - 0.447i$
Analytic cond. $172.457$
Root an. cond. $13.1322$
Motivic weight $33$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.71e5i·2-s + 3.66e7i·3-s − 2.07e10·4-s + 6.27e12·6-s − 4.77e13i·7-s + 2.09e15i·8-s + 4.21e15·9-s − 1.37e17·11-s − 7.60e17i·12-s + 3.30e18i·13-s − 8.18e18·14-s + 1.79e20·16-s + 1.37e20i·17-s − 7.23e20i·18-s − 8.26e20·19-s + ⋯
L(s)  = 1  − 1.84i·2-s + 0.490i·3-s − 2.41·4-s + 0.907·6-s − 0.543i·7-s + 2.62i·8-s + 0.758·9-s − 0.903·11-s − 1.18i·12-s + 1.37i·13-s − 1.00·14-s + 2.43·16-s + 0.684i·17-s − 1.40i·18-s − 0.657·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(172.457\)
Root analytic conductor: \(13.1322\)
Motivic weight: \(33\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :33/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(17)\) \(\approx\) \(0.5861913074\)
\(L(\frac12)\) \(\approx\) \(0.5861913074\)
\(L(\frac{35}{2})\) 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 \( 1 + 1.71e5iT - 8.58e9T^{2} \)
3 \( 1 - 3.66e7iT - 5.55e15T^{2} \)
7 \( 1 + 4.77e13iT - 7.73e27T^{2} \)
11 \( 1 + 1.37e17T + 2.32e34T^{2} \)
13 \( 1 - 3.30e18iT - 5.75e36T^{2} \)
17 \( 1 - 1.37e20iT - 4.02e40T^{2} \)
19 \( 1 + 8.26e20T + 1.58e42T^{2} \)
23 \( 1 - 4.90e22iT - 8.65e44T^{2} \)
29 \( 1 + 2.29e24T + 1.81e48T^{2} \)
31 \( 1 + 1.68e24T + 1.64e49T^{2} \)
37 \( 1 + 6.34e25iT - 5.63e51T^{2} \)
41 \( 1 - 6.03e26T + 1.66e53T^{2} \)
43 \( 1 - 2.71e26iT - 8.02e53T^{2} \)
47 \( 1 + 6.74e27iT - 1.51e55T^{2} \)
53 \( 1 + 3.22e28iT - 7.96e56T^{2} \)
59 \( 1 + 4.27e28T + 2.74e58T^{2} \)
61 \( 1 - 4.00e29T + 8.23e58T^{2} \)
67 \( 1 - 1.15e30iT - 1.82e60T^{2} \)
71 \( 1 + 4.55e30T + 1.23e61T^{2} \)
73 \( 1 + 5.29e30iT - 3.08e61T^{2} \)
79 \( 1 + 1.47e31T + 4.18e62T^{2} \)
83 \( 1 - 7.50e30iT - 2.13e63T^{2} \)
89 \( 1 - 1.55e32T + 2.13e64T^{2} \)
97 \( 1 - 7.66e32iT - 3.65e65T^{2} \)
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

−10.70248522971878698674157681583, −9.825711118237843241751120558617, −8.986984265707617956900816467178, −7.40315158205305148097695361573, −5.34129708763976726528833207840, −4.09697773648268304552266590570, −3.73278047300118231801741616990, −2.16037392854743843752217168452, −1.49543293100702200577255695768, −0.15019502084543902654484933041, 0.76324417937013803222655298200, 2.58234336362141287627283829482, 4.31457619959493346867659814817, 5.36068513360452864997309803637, 6.21430224646597625883397178298, 7.38797954079403720516763307308, 8.002371011276628155749439599626, 9.140396470221274908967319458832, 10.45331907573981877577453757770, 12.68996741283310502036160746267

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