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Properties

Label 2-425-85.19-c1-0-20
Degree $2$
Conductor $425$
Sign $-0.921 - 0.387i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.292 + 0.292i)2-s + (−1 − 2.41i)3-s + 1.82i·4-s + (1 + 0.414i)6-s + (−1 − 0.414i)7-s + (−1.12 − 1.12i)8-s + (−2.70 + 2.70i)9-s + (−1 − 0.414i)11-s + (4.41 − 1.82i)12-s − 1.41·13-s + (0.414 − 0.171i)14-s − 3·16-s + (−3 + 2.82i)17-s − 1.58i·18-s + (−3.41 − 3.41i)19-s + ⋯
L(s)  = 1  + (−0.207 + 0.207i)2-s + (−0.577 − 1.39i)3-s + 0.914i·4-s + (0.408 + 0.169i)6-s + (−0.377 − 0.156i)7-s + (−0.396 − 0.396i)8-s + (−0.902 + 0.902i)9-s + (−0.301 − 0.124i)11-s + (1.27 − 0.527i)12-s − 0.392·13-s + (0.110 − 0.0458i)14-s − 0.750·16-s + (−0.727 + 0.685i)17-s − 0.373i·18-s + (−0.783 − 0.783i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.921 - 0.387i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.921 - 0.387i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (3 - 2.82i)T \)
good2 \( 1 + (0.292 - 0.292i)T - 2iT^{2} \)
3 \( 1 + (1 + 2.41i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (1 + 0.414i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1 + 0.414i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
19 \( 1 + (3.41 + 3.41i)T + 19iT^{2} \)
23 \( 1 + (1.58 - 3.82i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.70 - 4.12i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (3 - 1.24i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.46 - 3.53i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.12 - 7.53i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (3.41 + 3.41i)T + 43iT^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 + (-4.24 + 4.24i)T - 59iT^{2} \)
61 \( 1 + (-3.53 + 8.53i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 6.82iT - 67T^{2} \)
71 \( 1 + (-12.0 + 5i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-4.94 + 2.05i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-3.82 - 1.58i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-0.242 + 0.242i)T - 83iT^{2} \)
89 \( 1 + 9.41iT - 89T^{2} \)
97 \( 1 + (-5.94 + 2.46i)T + (68.5 - 68.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.00275154066812877708083506156, −9.650605503500711633510530476798, −8.461712094088775646650720759822, −7.84276778428535037952558298962, −6.74139466625076497267501010283, −6.51525938650821944907948057307, −4.96999559640665771319244717240, −3.42004887698769473364810613663, −2.01830063658283705206385857218, 0, 2.39289955090906246927740323385, 4.05241045658196203350523733790, 4.95124961730577736204645377171, 5.78602528569735038141163518109, 6.72645989616945193200399518182, 8.395170124393985841844533636386, 9.438369382461915981316549295987, 9.927280108828869419373807776074, 10.65062340727806412462916462046

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