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Properties

Label 2-832-208.181-c1-0-10
Degree $2$
Conductor $832$
Sign $0.0103 - 0.999i$
Analytic cond. $6.64355$
Root an. cond. $2.57750$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.15 + 2.15i)3-s + (0.707 − 0.707i)5-s − 0.223·7-s − 6.31i·9-s + (1.41 − 1.41i)11-s + (1.34 + 3.34i)13-s + 3.05i·15-s + 3·17-s + (4.46 + 4.46i)19-s + (0.483 − 0.483i)21-s − 6.31i·23-s + 4i·25-s + (7.15 + 7.15i)27-s + (0.316 − 0.316i)29-s − 4.24i·31-s + ⋯
L(s)  = 1  + (−1.24 + 1.24i)3-s + (0.316 − 0.316i)5-s − 0.0846·7-s − 2.10i·9-s + (0.426 − 0.426i)11-s + (0.373 + 0.927i)13-s + 0.788i·15-s + 0.727·17-s + (1.02 + 1.02i)19-s + (0.105 − 0.105i)21-s − 1.31i·23-s + 0.800i·25-s + (1.37 + 1.37i)27-s + (0.0587 − 0.0587i)29-s − 0.762i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0103 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0103 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.0103 - 0.999i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ 0.0103 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.744734 + 0.737060i\)
\(L(\frac12)\) \(\approx\) \(0.744734 + 0.737060i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-1.34 - 3.34i)T \)
good3 \( 1 + (2.15 - 2.15i)T - 3iT^{2} \)
5 \( 1 + (-0.707 + 0.707i)T - 5iT^{2} \)
7 \( 1 + 0.223T + 7T^{2} \)
11 \( 1 + (-1.41 + 1.41i)T - 11iT^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + (-4.46 - 4.46i)T + 19iT^{2} \)
23 \( 1 + 6.31iT - 23T^{2} \)
29 \( 1 + (-0.316 + 0.316i)T - 29iT^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 + (6.58 - 6.58i)T - 37iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (-5.47 - 5.47i)T + 43iT^{2} \)
47 \( 1 - 10.5iT - 47T^{2} \)
53 \( 1 + (-3.63 - 3.63i)T + 53iT^{2} \)
59 \( 1 + (7.74 - 7.74i)T - 59iT^{2} \)
61 \( 1 + (-4 + 4i)T - 61iT^{2} \)
67 \( 1 + (4.69 + 4.69i)T + 67iT^{2} \)
71 \( 1 - 0.671T + 71T^{2} \)
73 \( 1 - 3.79T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + (-11.9 - 11.9i)T + 83iT^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 - 8.48iT - 97T^{2} \)
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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

−10.48487221266454715320843876303, −9.556358020454650531507958397927, −9.221523746296570055027980412377, −7.957108664103937307117943761351, −6.53429905413934967786480192266, −5.95115713206359878993024502752, −5.11161159052904802594091141464, −4.25490053697503511689815446507, −3.34511545692806006211156433979, −1.19140353993496779957767347475, 0.73882947308169173288339686632, 1.94260315382173235131816008347, 3.38651929316354210598924308379, 5.14100053863742282645504772952, 5.61578504518669907092463270300, 6.61138198535383901310627246771, 7.21892560064785974859828155381, 7.953729110881248414097938918486, 9.220017351223195023747983694452, 10.29324062909346193378940504547

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