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Properties

Label 2-832-52.31-c1-0-9
Degree $2$
Conductor $832$
Sign $0.881 - 0.471i$
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  − 1.47i·3-s + (−0.707 + 0.707i)5-s + (−1.04 + 1.04i)7-s + 0.828·9-s + (−3.55 + 3.55i)11-s + (3.53 − 0.707i)13-s + (1.04 + 1.04i)15-s − 0.171i·17-s + (5.03 + 5.03i)19-s + (1.53 + 1.53i)21-s − 2.08·23-s + 4i·25-s − 5.64i·27-s + 3.41·29-s + (2.08 + 2.08i)31-s + ⋯
L(s)  = 1  − 0.850i·3-s + (−0.316 + 0.316i)5-s + (−0.393 + 0.393i)7-s + 0.276·9-s + (−1.07 + 1.07i)11-s + (0.980 − 0.196i)13-s + (0.269 + 0.269i)15-s − 0.0416i·17-s + (1.15 + 1.15i)19-s + (0.335 + 0.335i)21-s − 0.434·23-s + 0.800i·25-s − 1.08i·27-s + 0.634·29-s + (0.374 + 0.374i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ 0.881 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30919 + 0.328300i\)
\(L(\frac12)\) \(\approx\) \(1.30919 + 0.328300i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.53 + 0.707i)T \)
good3 \( 1 + 1.47iT - 3T^{2} \)
5 \( 1 + (0.707 - 0.707i)T - 5iT^{2} \)
7 \( 1 + (1.04 - 1.04i)T - 7iT^{2} \)
11 \( 1 + (3.55 - 3.55i)T - 11iT^{2} \)
17 \( 1 + 0.171iT - 17T^{2} \)
19 \( 1 + (-5.03 - 5.03i)T + 19iT^{2} \)
23 \( 1 + 2.08T + 23T^{2} \)
29 \( 1 - 3.41T + 29T^{2} \)
31 \( 1 + (-2.08 - 2.08i)T + 31iT^{2} \)
37 \( 1 + (-6.12 - 6.12i)T + 37iT^{2} \)
41 \( 1 + (-4.24 + 4.24i)T - 41iT^{2} \)
43 \( 1 + 5.64T + 43T^{2} \)
47 \( 1 + (-1.04 + 1.04i)T - 47iT^{2} \)
53 \( 1 + 8.24T + 53T^{2} \)
59 \( 1 + (-7.11 + 7.11i)T - 59iT^{2} \)
61 \( 1 - 4.82T + 61T^{2} \)
67 \( 1 + (1.47 + 1.47i)T + 67iT^{2} \)
71 \( 1 + (-9.02 - 9.02i)T + 71iT^{2} \)
73 \( 1 + (-1.65 - 1.65i)T + 73iT^{2} \)
79 \( 1 + 0.863iT - 79T^{2} \)
83 \( 1 + (0.610 + 0.610i)T + 83iT^{2} \)
89 \( 1 + (2.24 + 2.24i)T + 89iT^{2} \)
97 \( 1 + (2.41 - 2.41i)T - 97iT^{2} \)
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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

−10.14052342364370707332492512290, −9.643703205554598828072255357894, −8.237481557345156356738141955919, −7.73994442864882295885885141831, −6.91893171550942123909280124527, −6.06920280802547714691427361801, −5.05205307983955929124441942063, −3.74227414326901912334319541383, −2.62415173899969209789020338004, −1.36180530640191506466111083957, 0.75014056365933085516833899242, 2.84923292994166612979463183490, 3.80015399406754863604105346965, 4.65781508407330260037374566072, 5.61155688157146186032904817694, 6.63212216077430953258261068520, 7.75862948073637922796884725592, 8.499316179193313769818383758422, 9.421226320624815887107267752750, 10.13203858708503904897115800565

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