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Properties

Label 2-832-13.4-c1-0-3
Degree $2$
Conductor $832$
Sign $-0.252 - 0.967i$
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  + (0.866 − 1.5i)3-s + 3.46i·5-s + (−2.59 + 1.5i)7-s + (−4.33 − 2.5i)11-s + (1 − 3.46i)13-s + (5.19 + 2.99i)15-s + (3.5 + 6.06i)17-s + (−4.33 + 2.5i)19-s + 5.19i·21-s + (−2.59 + 4.5i)23-s − 6.99·25-s + 5.19·27-s + (−2.5 + 4.33i)29-s + 2i·31-s + (−7.5 + 4.33i)33-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)3-s + 1.54i·5-s + (−0.981 + 0.566i)7-s + (−1.30 − 0.753i)11-s + (0.277 − 0.960i)13-s + (1.34 + 0.774i)15-s + (0.848 + 1.47i)17-s + (−0.993 + 0.573i)19-s + 1.13i·21-s + (−0.541 + 0.938i)23-s − 1.39·25-s + 1.00·27-s + (−0.464 + 0.804i)29-s + 0.359i·31-s + (−1.30 + 0.753i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $-0.252 - 0.967i$
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: \((2,\ 832,\ (\ :1/2),\ -0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.644801 + 0.834766i\)
\(L(\frac12)\) \(\approx\) \(0.644801 + 0.834766i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 3.46i)T \)
good3 \( 1 + (-0.866 + 1.5i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 + (2.59 - 1.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.33 + 2.5i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.33 - 2.5i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.59 - 4.5i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + (4.5 + 2.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.59 - 4.5i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (-6.06 + 3.5i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.06 + 3.5i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.46iT - 73T^{2} \)
79 \( 1 - 3.46T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + (-1.5 - 0.866i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.5 + 4.33i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.60195774873160405284938116000, −9.825858627751316530929121688513, −8.376585893064068134046467771425, −7.943818392258182514897943923733, −7.08719998194854200612340886851, −6.12544295282820725141904832160, −5.64951691037786993164926234045, −3.46701889614255901547158827532, −3.03507686231489644484628742773, −1.97046832394155394436606828951, 0.45712327582775890214975589543, 2.37212028449560547990958365689, 3.73383518618122464812277916686, 4.53977221518602306500339049214, 5.14918830804095099354067444517, 6.51816711380605981272777082346, 7.52750766774822474121386055996, 8.565284769052063164123669039898, 9.207234192835200423614294617761, 9.877998204908862386110032871315

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