↑

Properties

Label 2-5e4-25.11-c1-0-4
Degree $2$
Conductor $625$
Sign $-0.944 + 0.328i$
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  + (0.405 − 0.294i)2-s + (−0.955 + 2.94i)3-s + (−0.540 + 1.66i)4-s + (0.478 + 1.47i)6-s − 0.0237·7-s + (0.580 + 1.78i)8-s + (−5.31 − 3.86i)9-s + (−2.89 + 2.10i)11-s + (−4.37 − 3.17i)12-s + (3.05 + 2.21i)13-s + (−0.00964 + 0.00700i)14-s + (−2.06 − 1.50i)16-s + (1.11 + 3.44i)17-s − 3.29·18-s + (−0.751 − 2.31i)19-s + ⋯
L(s)  = 1  + (0.286 − 0.208i)2-s + (−0.551 + 1.69i)3-s + (−0.270 + 0.831i)4-s + (0.195 + 0.601i)6-s − 0.00899·7-s + (0.205 + 0.631i)8-s + (−1.77 − 1.28i)9-s + (−0.874 + 0.635i)11-s + (−1.26 − 0.917i)12-s + (0.847 + 0.615i)13-s + (−0.00257 + 0.00187i)14-s + (−0.517 − 0.375i)16-s + (0.271 + 0.835i)17-s − 0.775·18-s + (−0.172 − 0.530i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-0.944 + 0.328i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ -0.944 + 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.151832 - 0.897689i\)
\(L(\frac12)\) \(\approx\) \(0.151832 - 0.897689i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.405 + 0.294i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.955 - 2.94i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 0.0237T + 7T^{2} \)
11 \( 1 + (2.89 - 2.10i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.05 - 2.21i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.11 - 3.44i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.751 + 2.31i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.38 + 1.00i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.19 + 3.66i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.85 + 5.71i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.298 + 0.217i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.31 - 4.59i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 0.174T + 43T^{2} \)
47 \( 1 + (2.41 - 7.42i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.77 + 8.53i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.60 - 2.61i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (7.45 - 5.41i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.38 - 4.25i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.99 - 9.22i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.19 - 2.32i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.99 - 9.20i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.76 - 8.51i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-13.7 + 10.0i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.854 - 2.62i)T + (-78.4 - 57.0i)T^{2} \)
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

−11.15466196502387397556518724180, −10.28465241861977893295658279857, −9.479419198948680673198782942180, −8.681913476451831582988855590958, −7.78636078979858491930530382546, −6.32568114139759281928456124204, −5.27698167631637594881939408090, −4.40382280799066907807179552024, −3.85942901794295263414111793853, −2.67357861739520735130646561124, 0.49637319074180791366100491298, 1.62442298547384013015034431777, 3.15143351667160358404025628760, 5.06352964253551707424056629354, 5.69403631711650749524878635962, 6.40240787709102235287877682420, 7.30372002881038812417017239458, 8.122817809259675078787994634925, 9.068678738515737207774771650487, 10.51925809041635046714273268154

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