↑

Properties

Label 2-425-5.4-c1-0-7
Degree $2$
Conductor $425$
Sign $-0.447 - 0.894i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
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  + i·2-s + i·3-s + 4-s − 6-s + i·7-s + 3i·8-s + 2·9-s − 4·11-s + i·12-s + i·13-s − 14-s − 16-s + i·17-s + 2i·18-s + 6·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s + 0.5·4-s − 0.408·6-s + 0.377i·7-s + 1.06i·8-s + 0.666·9-s − 1.20·11-s + 0.288i·12-s + 0.277i·13-s − 0.267·14-s − 0.250·16-s + 0.242i·17-s + 0.471i·18-s + 1.37·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.844885 + 1.36705i\)
\(L(\frac12)\) \(\approx\) \(0.844885 + 1.36705i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 - iT \)
good2 \( 1 - iT - 2T^{2} \)
3 \( 1 - iT - 3T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - iT - 13T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 11iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 7T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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

−11.32747316071937976220647828529, −10.53977168770839914671963012133, −9.699677461183520456472066567750, −8.630823289347306924389441892267, −7.60084711130629951997117050429, −6.96919964975090527367086605617, −5.62190487616811240677112573918, −5.05764505512543381811566969493, −3.52247902799370925485161591156, −2.11470305054087019360518800532, 1.10409486004601865020106532370, 2.44349194376163007043486725423, 3.55529778655264588943773253420, 5.01455878607772551199993160919, 6.28447827990415011460858667792, 7.42938471590874538525234107269, 7.70435848304743941488685137839, 9.375267854060322023187195627850, 10.22430697620942944638591880988, 10.86272911899956796142579682521

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