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Properties

Label 2-832-52.47-c1-0-16
Degree $2$
Conductor $832$
Sign $0.289 + 0.957i$
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  − 2i·3-s + (1 + i)5-s + (1 + i)7-s − 9-s + (−3 − 3i)11-s + (3 − 2i)13-s + (2 − 2i)15-s + 4i·17-s + (3 − 3i)19-s + (2 − 2i)21-s − 3i·25-s − 4i·27-s + 6·29-s + (3 − 3i)31-s + (−6 + 6i)33-s + ⋯
L(s)  = 1  − 1.15i·3-s + (0.447 + 0.447i)5-s + (0.377 + 0.377i)7-s − 0.333·9-s + (−0.904 − 0.904i)11-s + (0.832 − 0.554i)13-s + (0.516 − 0.516i)15-s + 0.970i·17-s + (0.688 − 0.688i)19-s + (0.436 − 0.436i)21-s − 0.600i·25-s − 0.769i·27-s + 1.11·29-s + (0.538 − 0.538i)31-s + (−1.04 + 1.04i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.289 + 0.957i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ 0.289 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41474 - 1.04981i\)
\(L(\frac12)\) \(\approx\) \(1.41474 - 1.04981i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 + 2i)T \)
good3 \( 1 + 2iT - 3T^{2} \)
5 \( 1 + (-1 - i)T + 5iT^{2} \)
7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 + (3 + 3i)T + 11iT^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + (-3 + 3i)T - 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-3 + 3i)T - 31iT^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + (-1 - i)T + 41iT^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-5 - 5i)T + 47iT^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (7 + 7i)T + 59iT^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + (5 - 5i)T - 67iT^{2} \)
71 \( 1 + (5 - 5i)T - 71iT^{2} \)
73 \( 1 + (-9 + 9i)T - 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (-7 + 7i)T - 83iT^{2} \)
89 \( 1 + (-5 + 5i)T - 89iT^{2} \)
97 \( 1 + (-13 - 13i)T + 97iT^{2} \)
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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

−10.30801659593574229906062662854, −8.977294528658243183457779464505, −8.115116771733823337233228199522, −7.65519989075122318870640744523, −6.30171558880389432113683100010, −6.08714921890455553834316998089, −4.83862573600199225412673033765, −3.21450303942935321834060473074, −2.28929340995608555660815452645, −0.987897155123466237419653650317, 1.49646971882998607716059421516, 3.05865561672862757605365514436, 4.28856249206481940686889402602, 4.87659349413784554814133071281, 5.69598506309602603334017029567, 7.03638831355132460886714924305, 7.88186734248441434284994940545, 9.034423608864720248074139689161, 9.530967967390416147191781270793, 10.38704715435324824411506073762

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