↑

Properties

Label 2-925-185.99-c1-0-24
Degree $2$
Conductor $925$
Sign $0.834 - 0.550i$
Analytic cond. $7.38616$
Root an. cond. $2.71774$
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  + (−1.93 + 1.61i)2-s + (0.812 − 0.968i)3-s + (0.755 − 4.28i)4-s + 3.18i·6-s + (−1.09 − 2.99i)7-s + (2.95 + 5.12i)8-s + (0.243 + 1.37i)9-s + (2.73 + 4.73i)11-s + (−3.53 − 4.21i)12-s + (0.265 − 1.50i)13-s + (6.96 + 4.02i)14-s + (−5.83 − 2.12i)16-s + (0.140 + 0.794i)17-s + (−2.70 − 2.26i)18-s + (1.35 − 1.61i)19-s + ⋯
L(s)  = 1  + (−1.36 + 1.14i)2-s + (0.469 − 0.559i)3-s + (0.377 − 2.14i)4-s + 1.30i·6-s + (−0.412 − 1.13i)7-s + (1.04 + 1.81i)8-s + (0.0810 + 0.459i)9-s + (0.823 + 1.42i)11-s + (−1.02 − 1.21i)12-s + (0.0737 − 0.418i)13-s + (1.86 + 1.07i)14-s + (−1.45 − 0.531i)16-s + (0.0339 + 0.192i)17-s + (−0.637 − 0.534i)18-s + (0.311 − 0.371i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.834 - 0.550i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.834 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.883079 + 0.264864i\)
\(L(\frac12)\) \(\approx\) \(0.883079 + 0.264864i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 + (-6.05 + 0.539i)T \)
good2 \( 1 + (1.93 - 1.61i)T + (0.347 - 1.96i)T^{2} \)
3 \( 1 + (-0.812 + 0.968i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (1.09 + 2.99i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (-2.73 - 4.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.265 + 1.50i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.140 - 0.794i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (-1.35 + 1.61i)T + (-3.29 - 18.7i)T^{2} \)
23 \( 1 + (1.74 - 3.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.122 + 0.0705i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.37iT - 31T^{2} \)
41 \( 1 + (-0.0732 + 0.415i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 - 2.00T + 43T^{2} \)
47 \( 1 + (-1.46 - 0.842i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.49 + 6.86i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-3.65 + 10.0i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (-11.3 - 2.00i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-2.64 - 7.25i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-6.78 - 5.69i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + 8.77iT - 73T^{2} \)
79 \( 1 + (1.54 + 4.24i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-10.3 + 1.81i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (0.914 - 2.51i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (7.71 - 13.3i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.862884343269597932896636843166, −9.289857872087381546426892083510, −8.211768627826942207475802431475, −7.58050752850767088272116404067, −7.07352384302839311837758346734, −6.44083845748999125764059464680, −5.17875149274047464997885913318, −3.93877455117518568948404694181, −2.08898061537310024825385066805, −0.940070185996905051640105630171, 0.946818580526386215706841759045, 2.46389190647269335325308847300, 3.25072779511518275591757178049, 4.04346231667672313228586626656, 5.79897433302820830026688931996, 6.73062720899331136030663213058, 8.123591749804827434487242629891, 8.750155692088469257973527959702, 9.193052993568634475460030424845, 9.773244141251079029811230288034

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