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Properties

Label 2-832-52.47-c1-0-23
Degree $2$
Conductor $832$
Sign $-0.881 - 0.471i$
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  − 2.79i·3-s + (0.707 + 0.707i)5-s + (−1.97 − 1.97i)7-s − 4.82·9-s + (−1.15 − 1.15i)11-s + (−3.53 − 0.707i)13-s + (1.97 − 1.97i)15-s + 5.82i·17-s + (−1.63 + 1.63i)19-s + (−5.53 + 5.53i)21-s − 3.95·23-s − 4i·25-s + 5.11i·27-s + 0.585·29-s + (3.95 − 3.95i)31-s + ⋯
L(s)  = 1  − 1.61i·3-s + (0.316 + 0.316i)5-s + (−0.747 − 0.747i)7-s − 1.60·9-s + (−0.349 − 0.349i)11-s + (−0.980 − 0.196i)13-s + (0.510 − 0.510i)15-s + 1.41i·17-s + (−0.376 + 0.376i)19-s + (−1.20 + 1.20i)21-s − 0.825·23-s − 0.800i·25-s + 0.984i·27-s + 0.108·29-s + (0.710 − 0.710i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $-0.881 - 0.471i$
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: \((2,\ 832,\ (\ :1/2),\ -0.881 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.156590 + 0.624452i\)
\(L(\frac12)\) \(\approx\) \(0.156590 + 0.624452i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.53 + 0.707i)T \)
good3 \( 1 + 2.79iT - 3T^{2} \)
5 \( 1 + (-0.707 - 0.707i)T + 5iT^{2} \)
7 \( 1 + (1.97 + 1.97i)T + 7iT^{2} \)
11 \( 1 + (1.15 + 1.15i)T + 11iT^{2} \)
17 \( 1 - 5.82iT - 17T^{2} \)
19 \( 1 + (1.63 - 1.63i)T - 19iT^{2} \)
23 \( 1 + 3.95T + 23T^{2} \)
29 \( 1 - 0.585T + 29T^{2} \)
31 \( 1 + (-3.95 + 3.95i)T - 31iT^{2} \)
37 \( 1 + (-1.87 + 1.87i)T - 37iT^{2} \)
41 \( 1 + (4.24 + 4.24i)T + 41iT^{2} \)
43 \( 1 + 5.11T + 43T^{2} \)
47 \( 1 + (-1.97 - 1.97i)T + 47iT^{2} \)
53 \( 1 - 0.242T + 53T^{2} \)
59 \( 1 + (-2.31 - 2.31i)T + 59iT^{2} \)
61 \( 1 + 0.828T + 61T^{2} \)
67 \( 1 + (-2.79 + 2.79i)T - 67iT^{2} \)
71 \( 1 + (5.25 - 5.25i)T - 71iT^{2} \)
73 \( 1 + (9.65 - 9.65i)T - 73iT^{2} \)
79 \( 1 + 9.55iT - 79T^{2} \)
83 \( 1 + (6.75 - 6.75i)T - 83iT^{2} \)
89 \( 1 + (-6.24 + 6.24i)T - 89iT^{2} \)
97 \( 1 + (-0.414 - 0.414i)T + 97iT^{2} \)
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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

−10.03289551943845192124042706261, −8.535652550281100975533150414345, −7.891031697241568629432037349432, −7.08996850275575455138508461534, −6.35723832516745330083307404362, −5.76698897986102220034456523943, −4.11490820980019289112985037922, −2.81595902846700566324073422013, −1.83509022863634520125258108749, −0.29154209070280056762651201965, 2.47212299169159612768651097718, 3.33853173270043554407587153831, 4.71714689637399333929828978614, 5.05106218269654463357466806780, 6.11532833460491212666857296483, 7.26151081563176313301855819938, 8.570099782217173949096027606081, 9.309257995555090647888265855070, 9.789962934227032737223314629186, 10.32755614596692000996799873458

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