↑

Properties

Label 2-425-85.49-c1-0-0
Degree $2$
Conductor $425$
Sign $-0.0642 + 0.997i$
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  + (−0.680 + 0.680i)2-s + (−2.44 + 1.01i)3-s + 1.07i·4-s + (0.976 − 2.35i)6-s + (−1.18 + 2.85i)7-s + (−2.09 − 2.09i)8-s + (2.84 − 2.84i)9-s + (−2.34 + 5.66i)11-s + (−1.08 − 2.62i)12-s − 1.16·13-s + (−1.14 − 2.75i)14-s + 0.703·16-s + (3.92 + 1.25i)17-s + 3.86i·18-s + (−3.83 − 3.83i)19-s + ⋯
L(s)  = 1  + (−0.481 + 0.481i)2-s + (−1.41 + 0.585i)3-s + 0.536i·4-s + (0.398 − 0.962i)6-s + (−0.447 + 1.08i)7-s + (−0.739 − 0.739i)8-s + (0.946 − 0.946i)9-s + (−0.707 + 1.70i)11-s + (−0.313 − 0.757i)12-s − 0.321·13-s + (−0.304 − 0.735i)14-s + 0.175·16-s + (0.952 + 0.305i)17-s + 0.911i·18-s + (−0.880 − 0.880i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0642 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0642 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.0642 + 0.997i$
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: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.0642 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147122 - 0.156896i\)
\(L(\frac12)\) \(\approx\) \(0.147122 - 0.156896i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.92 - 1.25i)T \)
good2 \( 1 + (0.680 - 0.680i)T - 2iT^{2} \)
3 \( 1 + (2.44 - 1.01i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (1.18 - 2.85i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (2.34 - 5.66i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 1.16T + 13T^{2} \)
19 \( 1 + (3.83 + 3.83i)T + 19iT^{2} \)
23 \( 1 + (-2.88 - 1.19i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-4.61 + 1.91i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (1.42 + 3.44i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-0.366 + 0.151i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.57 - 0.651i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (0.0189 + 0.0189i)T + 43iT^{2} \)
47 \( 1 + 5.43T + 47T^{2} \)
53 \( 1 + (0.244 - 0.244i)T - 53iT^{2} \)
59 \( 1 + (2.87 - 2.87i)T - 59iT^{2} \)
61 \( 1 + (11.4 + 4.76i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 5.62iT - 67T^{2} \)
71 \( 1 + (-4.12 - 9.95i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-0.633 - 1.52i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (2.01 - 4.87i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (8.78 - 8.78i)T - 83iT^{2} \)
89 \( 1 - 3.22iT - 89T^{2} \)
97 \( 1 + (-4.94 - 11.9i)T + (-68.5 + 68.5i)T^{2} \)
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

−11.99939877892759147752339758852, −10.89221774772859140569886168778, −9.868815866751631410444064673821, −9.379242837756175321149014134582, −8.131968240596411246776415427447, −7.08320788681242119434728439165, −6.25760067549741563842207896831, −5.25812830796020759421894833362, −4.35463659746212890430258218923, −2.66563620111819903492926860126, 0.21552704125838563884136930257, 1.19773990485450099828181312382, 3.16590983395583757878339734615, 4.95184098809944109413718045442, 5.86114036687418125923689874209, 6.49809200447005500504936393279, 7.65095437862172739647605733157, 8.764315342360293389088041365642, 10.17492632503960415117171321929, 10.55465777102475885250652205177

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