Properties

Label 2-825-825.362-c0-0-0
Degree $2$
Conductor $825$
Sign $0.992 + 0.125i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)3-s + (0.951 − 0.309i)4-s i·5-s + (−0.809 − 0.587i)9-s + (0.587 − 0.809i)11-s + 0.999i·12-s + (0.951 + 0.309i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)20-s + (0.142 + 0.896i)23-s − 25-s + (0.809 − 0.587i)27-s + (−0.363 + 1.11i)31-s + (0.587 + 0.809i)33-s + (−0.951 − 0.309i)36-s + (−1.39 − 0.221i)37-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s + (0.951 − 0.309i)4-s i·5-s + (−0.809 − 0.587i)9-s + (0.587 − 0.809i)11-s + 0.999i·12-s + (0.951 + 0.309i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)20-s + (0.142 + 0.896i)23-s − 25-s + (0.809 − 0.587i)27-s + (−0.363 + 1.11i)31-s + (0.587 + 0.809i)33-s + (−0.951 − 0.309i)36-s + (−1.39 − 0.221i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.992 + 0.125i$
Analytic conductor: \(0.411728\)
Root analytic conductor: \(0.641660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :0),\ 0.992 + 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.089126899\)
\(L(\frac12)\) \(\approx\) \(1.089126899\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 - 0.951i)T \)
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 \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \)
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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

−10.59227814340391312955607888100, −9.492347674451474681731865905667, −8.977913857058806899397862206976, −7.986207839475145601437741091499, −6.79453439821690544556875867796, −5.79427043309097799065621908848, −5.27048785127432422817149779983, −4.07655107467333665645779168471, −3.07231977782587371939538560948, −1.36128657962000550185012757146, 1.84040638763147932381634540696, 2.63800235887392112417624965074, 3.84313449862990803194203262643, 5.46218372585940788903651114760, 6.51603998813645552489762575762, 6.87448913179785316035369646957, 7.59810552999309300554720697261, 8.464077879772912950762184474772, 9.802765100732143452137575283824, 10.72773099085772642578918514390

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