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Properties

Label 2-425-85.49-c1-0-13
Degree $2$
Conductor $425$
Sign $0.405 + 0.914i$
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  + (−1.70 + 1.70i)2-s + (−1 + 0.414i)3-s − 3.82i·4-s + (1 − 2.41i)6-s + (−1 + 2.41i)7-s + (3.12 + 3.12i)8-s + (−1.29 + 1.29i)9-s + (−1 + 2.41i)11-s + (1.58 + 3.82i)12-s + 1.41·13-s + (−2.41 − 5.82i)14-s − 2.99·16-s + (−3 − 2.82i)17-s − 4.41i·18-s + (−0.585 − 0.585i)19-s + ⋯
L(s)  = 1  + (−1.20 + 1.20i)2-s + (−0.577 + 0.239i)3-s − 1.91i·4-s + (0.408 − 0.985i)6-s + (−0.377 + 0.912i)7-s + (1.10 + 1.10i)8-s + (−0.430 + 0.430i)9-s + (−0.301 + 0.727i)11-s + (0.457 + 1.10i)12-s + 0.392·13-s + (−0.645 − 1.55i)14-s − 0.749·16-s + (−0.727 − 0.685i)17-s − 1.04i·18-s + (−0.134 − 0.134i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.405 + 0.914i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.405 + 0.914i)\)

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 \( 1 + (1.70 - 1.70i)T - 2iT^{2} \)
3 \( 1 + (1 - 0.414i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (1 - 2.41i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (1 - 2.41i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
19 \( 1 + (0.585 + 0.585i)T + 19iT^{2} \)
23 \( 1 + (4.41 + 1.82i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-0.292 + 0.121i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (3 + 7.24i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-8.53 + 3.53i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.12 - 0.464i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (0.585 + 0.585i)T + 43iT^{2} \)
47 \( 1 + 5.17T + 47T^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 + (4.24 - 4.24i)T - 59iT^{2} \)
61 \( 1 + (3.53 + 1.46i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 1.17iT - 67T^{2} \)
71 \( 1 + (2.07 + 5i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (4.94 + 11.9i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (1.82 - 4.41i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (8.24 - 8.24i)T - 83iT^{2} \)
89 \( 1 + 6.58iT - 89T^{2} \)
97 \( 1 + (3.94 + 9.53i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.76105691297342942984990241090, −9.791903228302149355782756766804, −9.149216817348982186780817812810, −8.247876836264968698360555225332, −7.41365566151151161894465217570, −6.25003174254374457773934081906, −5.74969100367221134875452148429, −4.62157351961607601781014946113, −2.35485851093939593475873485339, 0, 1.33682807989344830247393465810, 3.01217169934508413766871457344, 3.99425172762489601557091065438, 5.83150170105886121986071238743, 6.83732561576051257122083323241, 8.043099558011723850337980177806, 8.755557183981584333661880105684, 9.744288060010937475913242608215, 10.61147723123329600119032834426

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