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Properties

Label 2-832-13.4-c1-0-11
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

\(L(1)\) \(\approx\) \(1.01275 + 1.31112i\)
\(L(\frac12)\) \(\approx\) \(1.01275 + 1.31112i\)
\(L(\frac{3}{2})\) 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

−10.37855538303046272351632864330, −10.14697976683077386098533373380, −8.875654501507909549055467828457, −7.66284171828866836188876963342, −7.10207356397291731865845986530, −6.07121318228199451308780861511, −5.07820939062462916332348453283, −4.05599997123739660320989240945, −3.29395637560105612513211680269, −1.63865794362224080110720347500, 1.09493626110578598642653348604, 1.54146021753320441311826664331, 3.60662304927020795835689457928, 4.85392431774195462267657271737, 5.48757560664145178406598866342, 6.40409613211875862825593521930, 7.48304179116093703173315492948, 8.248679383802748163259605513561, 9.269785943212156608188685962966, 9.441302178174483087967774271641

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