Properties

Label 2-5e2-25.12-c2-0-1
Degree $2$
Conductor $25$
Sign $0.952 - 0.305i$
Analytic cond. $0.681200$
Root an. cond. $0.825348$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0933 + 0.589i)2-s + (0.210 + 0.107i)3-s + (3.46 − 1.12i)4-s + (−3.31 + 3.74i)5-s + (−0.0436 + 0.134i)6-s + (−7.64 − 7.64i)7-s + (2.07 + 4.06i)8-s + (−5.25 − 7.23i)9-s + (−2.51 − 1.60i)10-s + (7.33 + 5.33i)11-s + (0.851 + 0.134i)12-s + (−2.12 + 13.4i)13-s + (3.79 − 5.21i)14-s + (−1.10 + 0.433i)15-s + (9.59 − 6.96i)16-s + (0.704 − 0.359i)17-s + ⋯
L(s)  = 1  + (0.0466 + 0.294i)2-s + (0.0702 + 0.0357i)3-s + (0.866 − 0.281i)4-s + (−0.662 + 0.748i)5-s + (−0.00726 + 0.0223i)6-s + (−1.09 − 1.09i)7-s + (0.258 + 0.508i)8-s + (−0.584 − 0.803i)9-s + (−0.251 − 0.160i)10-s + (0.667 + 0.484i)11-s + (0.0709 + 0.0112i)12-s + (−0.163 + 1.03i)13-s + (0.270 − 0.372i)14-s + (−0.0733 + 0.0288i)15-s + (0.599 − 0.435i)16-s + (0.0414 − 0.0211i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.952 - 0.305i$
Analytic conductor: \(0.681200\)
Root analytic conductor: \(0.825348\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :1),\ 0.952 - 0.305i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.961180 + 0.150197i\)
\(L(\frac12)\) \(\approx\) \(0.961180 + 0.150197i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (3.31 - 3.74i)T \)
good2 \( 1 + (-0.0933 - 0.589i)T + (-3.80 + 1.23i)T^{2} \)
3 \( 1 + (-0.210 - 0.107i)T + (5.29 + 7.28i)T^{2} \)
7 \( 1 + (7.64 + 7.64i)T + 49iT^{2} \)
11 \( 1 + (-7.33 - 5.33i)T + (37.3 + 115. i)T^{2} \)
13 \( 1 + (2.12 - 13.4i)T + (-160. - 52.2i)T^{2} \)
17 \( 1 + (-0.704 + 0.359i)T + (169. - 233. i)T^{2} \)
19 \( 1 + (-13.5 - 4.40i)T + (292. + 212. i)T^{2} \)
23 \( 1 + (-13.5 + 2.14i)T + (503. - 163. i)T^{2} \)
29 \( 1 + (36.6 - 11.9i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (2.87 - 8.84i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-9.13 - 1.44i)T + (1.30e3 + 423. i)T^{2} \)
41 \( 1 + (33.7 - 24.5i)T + (519. - 1.59e3i)T^{2} \)
43 \( 1 + (-49.4 + 49.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-9.65 + 18.9i)T + (-1.29e3 - 1.78e3i)T^{2} \)
53 \( 1 + (56.8 + 28.9i)T + (1.65e3 + 2.27e3i)T^{2} \)
59 \( 1 + (45.6 + 62.8i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-11.4 - 8.29i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + (46.8 - 23.8i)T + (2.63e3 - 3.63e3i)T^{2} \)
71 \( 1 + (0.846 + 2.60i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-60.7 + 9.62i)T + (5.06e3 - 1.64e3i)T^{2} \)
79 \( 1 + (6.18 - 2.01i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-50.9 - 100. i)T + (-4.04e3 + 5.57e3i)T^{2} \)
89 \( 1 + (-65.5 + 90.2i)T + (-2.44e3 - 7.53e3i)T^{2} \)
97 \( 1 + (58.6 - 115. i)T + (-5.53e3 - 7.61e3i)T^{2} \)
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

−17.10479296573722037881498859686, −16.20221033606882414873804597711, −14.97679077349777733994060915547, −14.09200552801791932925696121316, −12.11249956695271839020945151232, −11.01487077074917571094520712864, −9.590994363266218252503016200986, −7.24460917229126064318521257037, −6.52286825665631089703808696221, −3.53355627929342546950010103975, 3.08943004397070765562960858767, 5.78075400279801035007472740552, 7.74516441709470580307876965470, 9.205830289706026577852263834769, 11.12964112980578338822113454307, 12.18579257629727377375926392867, 13.13614519039271023043333356267, 15.22208196423847419876229724966, 16.09769666240733093696647589496, 16.96152979915188095756561386716

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