Properties

Label 2-5e2-25.6-c5-0-4
Degree $2$
Conductor $25$
Sign $0.716 + 0.697i$
Analytic cond. $4.00959$
Root an. cond. $2.00239$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.447 − 1.37i)2-s + (−17.2 + 12.5i)3-s + (24.1 − 17.5i)4-s + (14.3 − 54.0i)5-s + (24.9 + 18.1i)6-s + 250.·7-s + (−72.5 − 52.7i)8-s + (65.1 − 200. i)9-s + (−80.8 + 4.37i)10-s + (−170. − 523. i)11-s + (−196. + 605. i)12-s + (−92.7 + 285. i)13-s + (−112. − 345. i)14-s + (428. + 1.11e3i)15-s + (255. − 786. i)16-s + (637. + 463. i)17-s + ⋯
L(s)  = 1  + (−0.0791 − 0.243i)2-s + (−1.10 + 0.803i)3-s + (0.755 − 0.549i)4-s + (0.257 − 0.966i)5-s + (0.283 + 0.205i)6-s + 1.93·7-s + (−0.400 − 0.291i)8-s + (0.268 − 0.825i)9-s + (−0.255 + 0.0138i)10-s + (−0.423 − 1.30i)11-s + (−0.394 + 1.21i)12-s + (−0.152 + 0.468i)13-s + (−0.152 − 0.470i)14-s + (0.491 + 1.27i)15-s + (0.249 − 0.768i)16-s + (0.535 + 0.388i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(4.00959\)
Root analytic conductor: \(2.00239\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :5/2),\ 0.716 + 0.697i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.25258 - 0.508742i\)
\(L(\frac12)\) \(\approx\) \(1.25258 - 0.508742i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-14.3 + 54.0i)T \)
good2 \( 1 + (0.447 + 1.37i)T + (-25.8 + 18.8i)T^{2} \)
3 \( 1 + (17.2 - 12.5i)T + (75.0 - 231. i)T^{2} \)
7 \( 1 - 250.T + 1.68e4T^{2} \)
11 \( 1 + (170. + 523. i)T + (-1.30e5 + 9.46e4i)T^{2} \)
13 \( 1 + (92.7 - 285. i)T + (-3.00e5 - 2.18e5i)T^{2} \)
17 \( 1 + (-637. - 463. i)T + (4.38e5 + 1.35e6i)T^{2} \)
19 \( 1 + (-432. - 314. i)T + (7.65e5 + 2.35e6i)T^{2} \)
23 \( 1 + (-252. - 777. i)T + (-5.20e6 + 3.78e6i)T^{2} \)
29 \( 1 + (-810. + 589. i)T + (6.33e6 - 1.95e7i)T^{2} \)
31 \( 1 + (-1.56e3 - 1.13e3i)T + (8.84e6 + 2.72e7i)T^{2} \)
37 \( 1 + (4.13e3 - 1.27e4i)T + (-5.61e7 - 4.07e7i)T^{2} \)
41 \( 1 + (3.66e3 - 1.12e4i)T + (-9.37e7 - 6.80e7i)T^{2} \)
43 \( 1 + 7.12e3T + 1.47e8T^{2} \)
47 \( 1 + (-6.17e3 + 4.48e3i)T + (7.08e7 - 2.18e8i)T^{2} \)
53 \( 1 + (1.62e4 - 1.18e4i)T + (1.29e8 - 3.97e8i)T^{2} \)
59 \( 1 + (1.15e3 - 3.56e3i)T + (-5.78e8 - 4.20e8i)T^{2} \)
61 \( 1 + (-3.13e3 - 9.63e3i)T + (-6.83e8 + 4.96e8i)T^{2} \)
67 \( 1 + (-3.08e4 - 2.23e4i)T + (4.17e8 + 1.28e9i)T^{2} \)
71 \( 1 + (-2.10e4 + 1.53e4i)T + (5.57e8 - 1.71e9i)T^{2} \)
73 \( 1 + (-5.96e3 - 1.83e4i)T + (-1.67e9 + 1.21e9i)T^{2} \)
79 \( 1 + (4.74e4 - 3.44e4i)T + (9.50e8 - 2.92e9i)T^{2} \)
83 \( 1 + (6.77e4 + 4.92e4i)T + (1.21e9 + 3.74e9i)T^{2} \)
89 \( 1 + (5.73e3 + 1.76e4i)T + (-4.51e9 + 3.28e9i)T^{2} \)
97 \( 1 + (-6.62e3 + 4.81e3i)T + (2.65e9 - 8.16e9i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.52422631303919314385171894276, −15.43015390034273328913167366487, −14.02668585624773181157585693404, −11.83526853476199739243744288281, −11.26371500950213309028569016333, −10.14749890589703913478395785732, −8.282785129088882683848353688402, −5.76158406167369993110368408898, −4.87580889375628582802523843856, −1.26125642613287895015667030781, 2.01138158014840650943553252660, 5.36654809447624634175871023802, 7.02814051001044802145683586207, 7.76114363027864992110362545237, 10.65630876521358654899472989260, 11.53590191738343198993663118163, 12.45447581672987063768179014476, 14.37756783809630697648569004132, 15.42389589614970930386198174742, 17.26927592256948780019167222041

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