↑

Properties

Label 2-825-825.98-c0-0-1
Degree $2$
Conductor $825$
Sign $0.904 - 0.425i$
Analytic cond. $0.411728$
Root an. cond. $0.641660$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 + 0.809i)3-s + (0.951 + 0.309i)4-s − i·5-s + (−0.309 + 0.951i)9-s + (−0.587 − 0.809i)11-s + (0.309 + 0.951i)12-s + (0.809 − 0.587i)15-s + (0.809 + 0.587i)16-s + (0.309 − 0.951i)20-s + (−0.142 + 0.896i)23-s − 25-s + (−0.951 + 0.309i)27-s + (−0.363 − 1.11i)31-s + (0.309 − 0.951i)33-s + (−0.587 + 0.809i)36-s + (−1.39 + 0.221i)37-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)3-s + (0.951 + 0.309i)4-s − i·5-s + (−0.309 + 0.951i)9-s + (−0.587 − 0.809i)11-s + (0.309 + 0.951i)12-s + (0.809 − 0.587i)15-s + (0.809 + 0.587i)16-s + (0.309 − 0.951i)20-s + (−0.142 + 0.896i)23-s − 25-s + (−0.951 + 0.309i)27-s + (−0.363 − 1.11i)31-s + (0.309 − 0.951i)33-s + (−0.587 + 0.809i)36-s + (−1.39 + 0.221i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.904 - 0.425i$
Analytic conductor: \(0.411728\)
Root analytic conductor: \(0.641660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :0),\ 0.904 - 0.425i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.365823155\)
\(L(\frac12)\) \(\approx\) \(1.365823155\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 + iT \)
11 \( 1 + (0.587 + 0.809i)T \)
good2 \( 1 + (-0.951 - 0.309i)T^{2} \)
7 \( 1 + iT^{2} \)
13 \( 1 + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (1.39 - 0.221i)T + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (1.26 - 0.642i)T + (0.587 - 0.809i)T^{2} \)
53 \( 1 + (-0.142 - 0.278i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2} \)
71 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \)
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

−10.42880081867839254874980855919, −9.659203283924769535899760896939, −8.674239708853917232133348297741, −8.128095150090656512625975239923, −7.31240156198509216564354073552, −5.89537896704927779659402307848, −5.20165458486054417941425881843, −3.98108006998495170800895694048, −3.10445755912102943938735644638, −1.88668283927960958726098017672, 1.82748626233868128143772319094, 2.61240696908239073657883811540, 3.55476418602277200000368845685, 5.26277235325848572154282124155, 6.43814711551472704754752019693, 6.90890192477061136543251418322, 7.59349966260660653491982463731, 8.450332669139959906299783196623, 9.710015532686703630293816265898, 10.40928416704458976225872482549

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