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Properties

Label 2-832-13.11-c0-0-0
Degree $2$
Conductor $832$
Sign $0.846 - 0.533i$
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  + (1.36 + 1.36i)5-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)13-s + (−0.866 − 0.5i)17-s + 2.73i·25-s + (−0.866 − 1.5i)29-s + (0.5 + 0.133i)37-s + (−0.133 + 0.5i)41-s + (1.86 − 0.499i)45-s + (0.866 − 0.5i)49-s − 53-s + (0.5 − 0.866i)61-s + (−1.86 − 0.499i)65-s + (0.366 − 0.366i)73-s + (−0.499 − 0.866i)81-s + ⋯
L(s)  = 1  + (1.36 + 1.36i)5-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)13-s + (−0.866 − 0.5i)17-s + 2.73i·25-s + (−0.866 − 1.5i)29-s + (0.5 + 0.133i)37-s + (−0.133 + 0.5i)41-s + (1.86 − 0.499i)45-s + (0.866 − 0.5i)49-s − 53-s + (0.5 − 0.866i)61-s + (−1.86 − 0.499i)65-s + (0.366 − 0.366i)73-s + (−0.499 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.846 - 0.533i$
Analytic conductor: \(0.415222\)
Root analytic conductor: \(0.644377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :0),\ 0.846 - 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.188263981\)
\(L(\frac12)\) \(\approx\) \(1.188263981\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
7 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
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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

−10.37382200025768666036247918630, −9.466907280217285571772487614463, −9.406868570456088469415445568881, −7.71737650154475897965482258027, −6.76184675791351524618622552817, −6.42936583727978270483653683299, −5.35879540791534021062627480148, −4.06990739768071370138353787532, −2.81201065437481343476282945082, −1.96304313090749609697792757956, 1.52583611447171103101202627383, 2.44459611662714200620811490083, 4.30285106476210617640516640235, 5.11340395392256713975733595580, 5.69332215507483514729803568191, 6.86632471771616723842744582839, 7.927356117549501727336875750533, 8.812284877693319782250465107802, 9.459958437149121086492526454523, 10.24368090345367036235719720955

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