Properties

Label 2-5e2-25.14-c1-0-0
Degree $2$
Conductor $25$
Sign $0.930 - 0.367i$
Analytic cond. $0.199626$
Root an. cond. $0.446795$
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  + (0.666 + 0.917i)2-s + (−2.47 − 0.804i)3-s + (0.220 − 0.679i)4-s + (−1.07 + 1.95i)5-s + (−0.911 − 2.80i)6-s − 0.407i·7-s + (2.92 − 0.951i)8-s + (3.05 + 2.21i)9-s + (−2.51 + 0.316i)10-s + (−1.61 + 1.17i)11-s + (−1.09 + 1.50i)12-s + (−0.411 + 0.566i)13-s + (0.373 − 0.271i)14-s + (4.24 − 3.98i)15-s + (1.66 + 1.21i)16-s + (1.50 − 0.489i)17-s + ⋯
L(s)  = 1  + (0.471 + 0.648i)2-s + (−1.42 − 0.464i)3-s + (0.110 − 0.339i)4-s + (−0.482 + 0.876i)5-s + (−0.372 − 1.14i)6-s − 0.153i·7-s + (1.03 − 0.336i)8-s + (1.01 + 0.739i)9-s + (−0.795 + 0.100i)10-s + (−0.487 + 0.354i)11-s + (−0.315 + 0.434i)12-s + (−0.114 + 0.157i)13-s + (0.0998 − 0.0725i)14-s + (1.09 − 1.02i)15-s + (0.416 + 0.302i)16-s + (0.365 − 0.118i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.930 - 0.367i$
Analytic conductor: \(0.199626\)
Root analytic conductor: \(0.446795\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :1/2),\ 0.930 - 0.367i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.575651 + 0.109512i\)
\(L(\frac12)\) \(\approx\) \(0.575651 + 0.109512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.07 - 1.95i)T \)
good2 \( 1 + (-0.666 - 0.917i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (2.47 + 0.804i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + 0.407iT - 7T^{2} \)
11 \( 1 + (1.61 - 1.17i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.411 - 0.566i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.50 + 0.489i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.52 + 4.70i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.706 + 0.971i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.70 - 5.23i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.53 - 7.80i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.01 + 4.15i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.83 + 4.24i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 9.16iT - 43T^{2} \)
47 \( 1 + (-1.21 - 0.393i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.83 + 1.56i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.25 + 3.82i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-7.62 + 5.53i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.93 - 0.952i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.12 + 6.53i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.320 + 0.441i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.69 - 5.21i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.926 - 0.301i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (1.83 - 1.33i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (14.4 + 4.70i)T + (78.4 + 57.0i)T^{2} \)
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

−17.68296973844215146425621882962, −16.39686760930127014099007322947, −15.42380134169698420468346514145, −14.18247416548231549440801696337, −12.65708171670277493230060996613, −11.25052963837300223329317058998, −10.40517280847289381337226809466, −7.31387626590244159416471837223, −6.45852707771919679444432037317, −4.97705277057159681438592943029, 4.20351592571445873243353432084, 5.61736961642789035495983217468, 8.000658794287396241187871279826, 10.22112449742912841687069309850, 11.51741502475793758833805884454, 12.15808889334106640147332851855, 13.28822573171171614394016615363, 15.51773135190410166453081903842, 16.65516428398578302281301636836, 17.11270173042148761906637195050

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