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Properties

Label 2-832-52.35-c0-0-0
Degree $2$
Conductor $832$
Sign $0.964 + 0.265i$
Analytic cond. $0.415222$
Root an. cond. $0.644377$
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  + 5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)45-s + (−0.5 + 0.866i)49-s + 53-s + (−0.5 − 0.866i)61-s + (0.5 − 0.866i)65-s − 73-s + (−0.499 + 0.866i)81-s + (0.5 + 0.866i)85-s + ⋯
L(s)  = 1  + 5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)45-s + (−0.5 + 0.866i)49-s + 53-s + (−0.5 − 0.866i)61-s + (0.5 − 0.866i)65-s − 73-s + (−0.499 + 0.866i)81-s + (0.5 + 0.866i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.964 + 0.265i$
Analytic conductor: \(0.415222\)
Root analytic conductor: \(0.644377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :0),\ 0.964 + 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.128275941\)
\(L(\frac12)\) \(\approx\) \(1.128275941\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.34171195898141600290752045253, −9.550728528784383610325028863174, −8.788857535467445963694871471797, −7.960781945023319979958792070121, −6.74967866018315926630608153952, −5.91080782726032255117256853856, −5.38848199087459744727417017856, −3.85995080406162316945946703601, −2.90170479363272988772809371389, −1.45409004826711199249013006458, 1.76348168960243201443293740352, 2.75780103292694502802947627830, 4.18896647589809392843679974444, 5.33245819744144399996348455539, 5.94004397471897637449001737152, 6.99063060318122291659412215519, 7.930126170105022775739781557862, 8.903547049163200532714424180878, 9.605921739771089939772348901243, 10.37266678450107325880979581293

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