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Properties

Label 2-832-52.47-c1-0-17
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  + (−1 − i)5-s + 3·9-s + (−2 − 3i)13-s − 2i·17-s − 3i·25-s + 4·29-s + (7 − 7i)37-s + (−9 − 9i)41-s + (−3 − 3i)45-s − 7i·49-s + 14·53-s − 10·61-s + (−1 + 5i)65-s + (5 − 5i)73-s + 9·81-s + ⋯
L(s)  = 1  + (−0.447 − 0.447i)5-s + 9-s + (−0.554 − 0.832i)13-s − 0.485i·17-s − 0.600i·25-s + 0.742·29-s + (1.15 − 1.15i)37-s + (−1.40 − 1.40i)41-s + (−0.447 − 0.447i)45-s − i·49-s + 1.92·53-s − 1.28·61-s + (−0.124 + 0.620i)65-s + (0.585 − 0.585i)73-s + 81-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

\(L(1)\) \(\approx\) \(1.08050 - 0.801797i\)
\(L(\frac12)\) \(\approx\) \(1.08050 - 0.801797i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 3i)T \)
good3 \( 1 - 3T^{2} \)
5 \( 1 + (1 + i)T + 5iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 + 11iT^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 31iT^{2} \)
37 \( 1 + (-7 + 7i)T - 37iT^{2} \)
41 \( 1 + (9 + 9i)T + 41iT^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 14T + 53T^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71iT^{2} \)
73 \( 1 + (-5 + 5i)T - 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (13 - 13i)T - 89iT^{2} \)
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

−10.11081235931218808484775954777, −9.221472717260353864888676080485, −8.278059953066293564914083222283, −7.51141104987534234324747991228, −6.74123374497659023336279004524, −5.47727154638230615836880122122, −4.63288123618797833350163336783, −3.71539162375779761475972452083, −2.34251563079843641397516445589, −0.71060958710711567148670955373, 1.51604154383517791386100761892, 2.93339273192081698933226158224, 4.10628128650544631600482402891, 4.83401247899382478262749333376, 6.23347567113737762826082679742, 6.99816290123132793083542187750, 7.69919666610636119465184847023, 8.675822134031912651049438441443, 9.723464124329590588067082061597, 10.25373964146720309162317462401

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