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Properties

Label 2-832-13.4-c1-0-13
Degree $2$
Conductor $832$
Sign $0.964 + 0.265i$
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 − 1.73i)3-s + 1.73i·5-s + (−0.499 − 0.866i)9-s + (2.5 + 2.59i)13-s + (2.99 + 1.73i)15-s + (1.5 + 2.59i)17-s + (3 − 1.73i)19-s + (3 − 5.19i)23-s + 2.00·25-s + 4.00·27-s + (1.5 − 2.59i)29-s + 3.46i·31-s + (−7.5 − 4.33i)37-s + (7 − 1.73i)39-s + (−4.5 − 2.59i)41-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s + 0.774i·5-s + (−0.166 − 0.288i)9-s + (0.693 + 0.720i)13-s + (0.774 + 0.447i)15-s + (0.363 + 0.630i)17-s + (0.688 − 0.397i)19-s + (0.625 − 1.08i)23-s + 0.400·25-s + 0.769·27-s + (0.278 − 0.482i)29-s + 0.622i·31-s + (−1.23 − 0.711i)37-s + (1.12 − 0.277i)39-s + (−0.702 − 0.405i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.964 + 0.265i$
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: \((2,\ 832,\ (\ :1/2),\ 0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00259 - 0.270180i\)
\(L(\frac12)\) \(\approx\) \(2.00259 - 0.270180i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 2.59i)T \)
good3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.73iT - 5T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (7.5 + 4.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (6 - 3.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3 + 1.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 1.73i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (6 + 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.35156224761615169015313377590, −9.081748904585127415739442859446, −8.451177385438783730373633503685, −7.48945524821787918253428785385, −6.84870111365291030756401628935, −6.16346685631927636428370955012, −4.77595904803463638852110189098, −3.44600387721085178412118898592, −2.52519132039434657488003662700, −1.34757788576870210394749588045, 1.19342195581952098885891454876, 3.06558804326536764668254183560, 3.73821556874204723903753598097, 4.92680899819577653089241091085, 5.48585094412207632847449169395, 6.87255681828692222734207562669, 7.981178807168000502414143638320, 8.719798006643904438057059673349, 9.393142522121543241355094837957, 10.05947621800990910940676282125

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