↑

Properties

Label 2-425-17.15-c1-0-8
Degree $2$
Conductor $425$
Sign $-0.865 - 0.500i$
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  + (1.44 + 1.44i)2-s + (1.22 + 2.95i)3-s + 2.18i·4-s + (−2.50 + 6.03i)6-s + (−1.08 − 0.450i)7-s + (−0.263 + 0.263i)8-s + (−5.09 + 5.09i)9-s + (1.88 − 4.54i)11-s + (−6.44 + 2.66i)12-s − 2.46i·13-s + (−0.921 − 2.22i)14-s + 3.60·16-s + (1.36 + 3.89i)17-s − 14.7·18-s + (−1.44 − 1.44i)19-s + ⋯
L(s)  = 1  + (1.02 + 1.02i)2-s + (0.706 + 1.70i)3-s + 1.09i·4-s + (−1.02 + 2.46i)6-s + (−0.411 − 0.170i)7-s + (−0.0931 + 0.0931i)8-s + (−1.69 + 1.69i)9-s + (0.567 − 1.37i)11-s + (−1.85 + 0.770i)12-s − 0.683i·13-s + (−0.246 − 0.594i)14-s + 0.900·16-s + (0.330 + 0.943i)17-s − 3.47·18-s + (−0.331 − 0.331i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.865 - 0.500i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.865 - 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.720693 + 2.68468i\)
\(L(\frac12)\) \(\approx\) \(0.720693 + 2.68468i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 - 3.89i)T \)
good2 \( 1 + (-1.44 - 1.44i)T + 2iT^{2} \)
3 \( 1 + (-1.22 - 2.95i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (1.08 + 0.450i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.88 + 4.54i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 2.46iT - 13T^{2} \)
19 \( 1 + (1.44 + 1.44i)T + 19iT^{2} \)
23 \( 1 + (-0.0455 + 0.109i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.984 + 0.407i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.06 - 2.58i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (0.885 + 2.13i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (0.662 + 0.274i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (7.13 - 7.13i)T - 43iT^{2} \)
47 \( 1 + 3.39iT - 47T^{2} \)
53 \( 1 + (9.84 + 9.84i)T + 53iT^{2} \)
59 \( 1 + (-1.07 + 1.07i)T - 59iT^{2} \)
61 \( 1 + (-7.46 - 3.09i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 4.92T + 67T^{2} \)
71 \( 1 + (-2.53 - 6.13i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (3.38 - 1.40i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-3.13 + 7.55i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-4.30 - 4.30i)T + 83iT^{2} \)
89 \( 1 - 8.46iT - 89T^{2} \)
97 \( 1 + (-4.14 + 1.71i)T + (68.5 - 68.5i)T^{2} \)
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

−11.39409011757889565563471927983, −10.47776474744047829140186089088, −9.796082439600209822952305908938, −8.607119044540039674795495968373, −8.089590913903604310400722657537, −6.53211028505274154296178906759, −5.65087756316103138372964125429, −4.77552843660092664107534329591, −3.69605154241206494718871792462, −3.24282157514331024793459709928, 1.52242108558737184862374957078, 2.37092671784520788220590253380, 3.40207621309054308589602788246, 4.69000195351767975965409751268, 6.14973529802431092527167900603, 7.01494639686532002284453142717, 7.84188770662085448725282502026, 9.053169549797537822986290360272, 9.936280571477070134046173404169, 11.41137807646777322987400634004

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