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Properties

Label 2-832-13.4-c1-0-21
Degree $2$
Conductor $832$
Sign $-0.895 + 0.446i$
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  + (−1.13 + 1.96i)3-s − 3.99i·5-s + (0.981 − 0.566i)7-s + (−1.06 − 1.84i)9-s + (−0.981 − 0.566i)11-s + (−3.59 − 0.266i)13-s + (7.84 + 4.52i)15-s + (−0.5 − 0.866i)17-s + (−3.39 + 1.96i)19-s + 2.56i·21-s + (−4.59 + 7.96i)23-s − 10.9·25-s − 1.96·27-s + (−2.02 + 3.51i)29-s − 9.05i·31-s + ⋯
L(s)  = 1  + (−0.654 + 1.13i)3-s − 1.78i·5-s + (0.371 − 0.214i)7-s + (−0.355 − 0.615i)9-s + (−0.295 − 0.170i)11-s + (−0.997 − 0.0740i)13-s + (2.02 + 1.16i)15-s + (−0.121 − 0.210i)17-s + (−0.779 + 0.450i)19-s + 0.560i·21-s + (−0.958 + 1.66i)23-s − 2.19·25-s − 0.377·27-s + (−0.376 + 0.652i)29-s − 1.62i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $-0.895 + 0.446i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ -0.895 + 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0488835 - 0.207677i\)
\(L(\frac12)\) \(\approx\) \(0.0488835 - 0.207677i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.59 + 0.266i)T \)
good3 \( 1 + (1.13 - 1.96i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.99iT - 5T^{2} \)
7 \( 1 + (-0.981 + 0.566i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.981 + 0.566i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.39 - 1.96i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.59 - 7.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.02 - 3.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.05iT - 31T^{2} \)
37 \( 1 + (2.16 + 1.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.42 + 1.39i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.24 - 5.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.79iT - 47T^{2} \)
53 \( 1 - 8.92T + 53T^{2} \)
59 \( 1 + (-7.77 + 4.49i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.89 + 6.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.36 + 4.82i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.86 - 3.96i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.29iT - 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 - 5.73iT - 83T^{2} \)
89 \( 1 + (-12.1 - 7.01i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.70 - 4.44i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.713852152347112756907748688123, −9.307917451276004435246575169287, −8.234813512649756088111076142912, −7.53714439958119048920253532424, −5.80311143699301186345121947835, −5.29788308080208709186588246371, −4.52750773694219081416850962596, −3.87589369729132173357941324867, −1.82682994441042935999692026541, −0.10597681615621885245657765640, 2.03168274348223613501746682388, 2.73741550835237352039293393700, 4.27918425659581991965164717656, 5.63516114469575871167772920068, 6.50168567099011332860051071513, 6.99652198774928423043266685049, 7.65450836284052770372767564406, 8.690728356885653481513869023316, 10.25062276131043177052867371227, 10.46041913292995635121862841421

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