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Properties

Label 2-832-104.101-c1-0-14
Degree $2$
Conductor $832$
Sign $0.246 + 0.969i$
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 + 0.5i)3-s + (−3.12 − 1.80i)7-s + (−1 + 1.73i)9-s + (0.866 + 1.5i)11-s + 3.60·13-s + (1.5 − 2.59i)17-s + (0.866 − 1.5i)19-s + 3.60·21-s + (−3.12 − 5.40i)23-s − 5·25-s − 5i·27-s + (5.40 − 3.12i)29-s − 7.21i·31-s + (−1.5 − 0.866i)33-s + (−1.80 − 3.12i)37-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (−1.18 − 0.681i)7-s + (−0.333 + 0.577i)9-s + (0.261 + 0.452i)11-s + 1.00·13-s + (0.363 − 0.630i)17-s + (0.198 − 0.344i)19-s + 0.786·21-s + (−0.651 − 1.12i)23-s − 25-s − 0.962i·27-s + (1.00 − 0.579i)29-s − 1.29i·31-s + (−0.261 − 0.150i)33-s + (−0.296 − 0.513i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.246 + 0.969i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ 0.246 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.647408 - 0.503401i\)
\(L(\frac12)\) \(\approx\) \(0.647408 - 0.503401i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.60T \)
good3 \( 1 + (0.866 - 0.5i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 + (3.12 + 1.80i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.866 - 1.5i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.866 + 1.5i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.12 + 5.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.40 + 3.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.21iT - 31T^{2} \)
37 \( 1 + (1.80 + 3.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 12.4iT - 53T^{2} \)
59 \( 1 + (6.06 - 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.40 - 3.12i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.866 + 1.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.36 + 5.40i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.46iT - 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 - 6.92T + 83T^{2} \)
89 \( 1 + (-7.5 + 4.33i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.5 + 6.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.08184081410043997493743685798, −9.441044472061838420289932047170, −8.319228401403583064302418495719, −7.40463126333700147705892906733, −6.40432132134352495390680595765, −5.79999924713761962098103893153, −4.53795606217992001104377864814, −3.73048678992419198325353371622, −2.43662897409828315085239845378, −0.46437949603690812662274263339, 1.31874673548924120852853098846, 3.11270570997036703328696172643, 3.74543383087958232459972163570, 5.44963780273806398433555201717, 6.14415273924661188905352533206, 6.55451431333082299012183313936, 7.88520031332903512421831957486, 8.828249520190240156909224656180, 9.464698692939950995251815970769, 10.39855848153417982046192255960

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