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Properties

Label 2-832-52.47-c1-0-2
Degree $2$
Conductor $832$
Sign $-0.971 - 0.238i$
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  + 2.89i·3-s + (2.84 + 2.84i)5-s + (−0.193 − 0.193i)7-s − 5.40·9-s + (−3.65 − 3.65i)11-s + (0.193 + 3.60i)13-s + (−8.25 + 8.25i)15-s + 0.899i·17-s + (−0.753 + 0.753i)19-s + (0.560 − 0.560i)21-s − 1.89·23-s + 11.2i·25-s − 6.97i·27-s + 5.41·29-s + (3.24 − 3.24i)31-s + ⋯
L(s)  = 1  + 1.67i·3-s + (1.27 + 1.27i)5-s + (−0.0730 − 0.0730i)7-s − 1.80·9-s + (−1.10 − 1.10i)11-s + (0.0536 + 0.998i)13-s + (−2.13 + 2.13i)15-s + 0.218i·17-s + (−0.172 + 0.172i)19-s + (0.122 − 0.122i)21-s − 0.394·23-s + 2.24i·25-s − 1.34i·27-s + 1.00·29-s + (0.583 − 0.583i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $-0.971 - 0.238i$
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: \((2,\ 832,\ (\ :1/2),\ -0.971 - 0.238i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.190702 + 1.57914i\)
\(L(\frac12)\) \(\approx\) \(0.190702 + 1.57914i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.193 - 3.60i)T \)
good3 \( 1 - 2.89iT - 3T^{2} \)
5 \( 1 + (-2.84 - 2.84i)T + 5iT^{2} \)
7 \( 1 + (0.193 + 0.193i)T + 7iT^{2} \)
11 \( 1 + (3.65 + 3.65i)T + 11iT^{2} \)
17 \( 1 - 0.899iT - 17T^{2} \)
19 \( 1 + (0.753 - 0.753i)T - 19iT^{2} \)
23 \( 1 + 1.89T + 23T^{2} \)
29 \( 1 - 5.41T + 29T^{2} \)
31 \( 1 + (-3.24 + 3.24i)T - 31iT^{2} \)
37 \( 1 + (2.95 - 2.95i)T - 37iT^{2} \)
41 \( 1 + (5.79 + 5.79i)T + 41iT^{2} \)
43 \( 1 - 2.81T + 43T^{2} \)
47 \( 1 + (-7.49 - 7.49i)T + 47iT^{2} \)
53 \( 1 - 5.69T + 53T^{2} \)
59 \( 1 + (-8.44 - 8.44i)T + 59iT^{2} \)
61 \( 1 - 2.98T + 61T^{2} \)
67 \( 1 + (8.85 - 8.85i)T - 67iT^{2} \)
71 \( 1 + (1.91 - 1.91i)T - 71iT^{2} \)
73 \( 1 + (-5.40 + 5.40i)T - 73iT^{2} \)
79 \( 1 + 1.20iT - 79T^{2} \)
83 \( 1 + (2.14 - 2.14i)T - 83iT^{2} \)
89 \( 1 + (1.79 - 1.79i)T - 89iT^{2} \)
97 \( 1 + (1.10 + 1.10i)T + 97iT^{2} \)
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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

−10.29937683135230644113818448182, −10.15487071877974680416001965384, −9.149102464314638251847288839868, −8.393180795502540380107916126657, −6.95870110915232536339854895100, −5.97284649978029734125711115673, −5.43892260081374406043006156370, −4.22237901371876902729000577099, −3.19224255015900919613112122631, −2.38792623303952068088442058220, 0.75915135847026632331050051444, 1.90291036839823282949415922712, 2.65190309040693480507406935423, 4.83478220168870915966733663328, 5.49641324875937337615546291188, 6.32351308192932507003730393146, 7.26893776571384930175546662712, 8.144460573827776824119009024306, 8.714303165795187704594351098083, 9.843039789551289723999196346120

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