↑

Properties

Label 2-5e4-25.6-c1-0-23
Degree $2$
Conductor $625$
Sign $0.699 - 0.714i$
Analytic cond. $4.99065$
Root an. cond. $2.23397$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.622 + 1.91i)2-s + (2.44 − 1.77i)3-s + (−1.66 + 1.20i)4-s + (4.92 + 3.58i)6-s − 0.369·7-s + (−0.0893 − 0.0649i)8-s + (1.90 − 5.85i)9-s + (−0.539 − 1.66i)11-s + (−1.92 + 5.91i)12-s + (−0.344 + 1.06i)13-s + (−0.230 − 0.708i)14-s + (−1.20 + 3.69i)16-s + (4.43 + 3.22i)17-s + 12.3·18-s + (3.03 + 2.20i)19-s + ⋯
L(s)  = 1  + (0.440 + 1.35i)2-s + (1.41 − 1.02i)3-s + (−0.831 + 0.603i)4-s + (2.01 + 1.46i)6-s − 0.139·7-s + (−0.0316 − 0.0229i)8-s + (0.633 − 1.95i)9-s + (−0.162 − 0.500i)11-s + (−0.554 + 1.70i)12-s + (−0.0956 + 0.294i)13-s + (−0.0615 − 0.189i)14-s + (−0.300 + 0.924i)16-s + (1.07 + 0.782i)17-s + 2.92·18-s + (0.696 + 0.505i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.699 - 0.714i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.699 - 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.71617 + 1.14177i\)
\(L(\frac12)\) \(\approx\) \(2.71617 + 1.14177i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.622 - 1.91i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-2.44 + 1.77i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 0.369T + 7T^{2} \)
11 \( 1 + (0.539 + 1.66i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.344 - 1.06i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.43 - 3.22i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.03 - 2.20i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.23 + 6.89i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.39 - 2.46i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.247 + 0.179i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.84 + 8.76i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.29 - 3.98i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.17T + 43T^{2} \)
47 \( 1 + (-0.655 + 0.476i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.16 - 2.30i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.573 - 1.76i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.99 - 9.21i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (10.0 + 7.32i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (9.44 - 6.86i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.06 + 3.26i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.60 - 3.34i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.77 - 4.19i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.472 - 1.45i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-5.17 + 3.76i)T + (29.9 - 92.2i)T^{2} \)
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

−10.54657660303997166258788019532, −9.385263062338484666694224456895, −8.453907018049965018192421762678, −7.958336240060816405475200149647, −7.26915598037539760117923627817, −6.41293868334716541888145063628, −5.61302523832300309456630317879, −4.12275948306166654352512664459, −3.07396848809243083185322745248, −1.66380560844967162105892649339, 1.76857592165036992731996127612, 3.02099834742824984667322725757, 3.39343311045635944743065282139, 4.51479558837542099111489688960, 5.29682680130681098064080507676, 7.32191716173418943657585900141, 8.026098062233334436075386067731, 9.317459651052304670512778925737, 9.772650005872771805118184360871, 10.23462971479288101112108791020

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