↑

Properties

Label 2-425-85.59-c1-0-11
Degree $2$
Conductor $425$
Sign $0.966 + 0.256i$
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  + (0.917 + 0.917i)2-s + (−1.69 − 0.703i)3-s − 0.318i·4-s + (−0.912 − 2.20i)6-s + (1.34 + 3.23i)7-s + (2.12 − 2.12i)8-s + (0.267 + 0.267i)9-s + (−1.46 − 3.53i)11-s + (−0.223 + 0.540i)12-s + 3.99·13-s + (−1.74 + 4.20i)14-s + 3.26·16-s + (2.51 − 3.26i)17-s + 0.490i·18-s + (3.98 − 3.98i)19-s + ⋯
L(s)  = 1  + (0.648 + 0.648i)2-s + (−0.980 − 0.406i)3-s − 0.159i·4-s + (−0.372 − 0.899i)6-s + (0.507 + 1.22i)7-s + (0.751 − 0.751i)8-s + (0.0890 + 0.0890i)9-s + (−0.441 − 1.06i)11-s + (−0.0645 + 0.155i)12-s + 1.10·13-s + (−0.465 + 1.12i)14-s + 0.815·16-s + (0.611 − 0.791i)17-s + 0.115i·18-s + (0.914 − 0.914i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.966 + 0.256i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.966 + 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51849 - 0.198371i\)
\(L(\frac12)\) \(\approx\) \(1.51849 - 0.198371i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.51 + 3.26i)T \)
good2 \( 1 + (-0.917 - 0.917i)T + 2iT^{2} \)
3 \( 1 + (1.69 + 0.703i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-1.34 - 3.23i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.46 + 3.53i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 3.99T + 13T^{2} \)
19 \( 1 + (-3.98 + 3.98i)T - 19iT^{2} \)
23 \( 1 + (2.31 - 0.960i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-5.14 - 2.12i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-0.843 + 2.03i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (1.78 + 0.738i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.730 + 0.302i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (0.704 - 0.704i)T - 43iT^{2} \)
47 \( 1 + 9.47T + 47T^{2} \)
53 \( 1 + (-3.87 - 3.87i)T + 53iT^{2} \)
59 \( 1 + (2.12 + 2.12i)T + 59iT^{2} \)
61 \( 1 + (12.9 - 5.37i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 15.9iT - 67T^{2} \)
71 \( 1 + (2.09 - 5.05i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-2.51 + 6.06i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-4.54 - 10.9i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (10.8 + 10.8i)T + 83iT^{2} \)
89 \( 1 + 4.92iT - 89T^{2} \)
97 \( 1 + (5.45 - 13.1i)T + (-68.5 - 68.5i)T^{2} \)
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

−11.44108254750162680389589301203, −10.51011601853258556472708642573, −9.210970143951580637856538420917, −8.258601369760602607877352529715, −7.06734631593927198039367865954, −6.01669727344474931532233097160, −5.64157435812079741264051642404, −4.86584022223957655553647999204, −3.11346120166255624884036300835, −1.06799412543329883342493773436, 1.57497881900485251946114063693, 3.43646001515816764822894419101, 4.37161194771167583268174266613, 5.08480855459943121032757760288, 6.25316027219527117612485628550, 7.62817438683023094113087115598, 8.212896208170197532738940381994, 10.09405987479042656674953283167, 10.47951175107490237945124433878, 11.24981115012732815385785236781

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