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Properties

Label 2-1792-56.27-c1-0-53
Degree $2$
Conductor $1792$
Sign $-0.0716 + 0.997i$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
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  − 2i·3-s + 3.46·5-s + (2 − 1.73i)7-s − 9-s − 3.46·11-s + 3.46·13-s − 6.92i·15-s − 2i·19-s + (−3.46 − 4i)21-s + 3.46i·23-s + 6.99·25-s − 4i·27-s − 6i·29-s − 8·31-s + 6.92i·33-s + ⋯
L(s)  = 1  − 1.15i·3-s + 1.54·5-s + (0.755 − 0.654i)7-s − 0.333·9-s − 1.04·11-s + 0.960·13-s − 1.78i·15-s − 0.458i·19-s + (−0.755 − 0.872i)21-s + 0.722i·23-s + 1.39·25-s − 0.769i·27-s − 1.11i·29-s − 1.43·31-s + 1.20i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0716 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0716 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.0716 + 0.997i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (895, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -0.0716 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.537093725\)
\(L(\frac12)\) \(\approx\) \(2.537093725\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good3 \( 1 + 2iT - 3T^{2} \)
5 \( 1 - 3.46T + 5T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 3.46T + 61T^{2} \)
67 \( 1 - 3.46T + 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 3.46iT - 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−9.084427067543730269907437332285, −8.045217027045244029392061587241, −7.53819140957394053979307362786, −6.69008259656637490972590677662, −5.88167872646949444854727333832, −5.31390039354101702332540263653, −4.12331770300204152270433532727, −2.62292988788677763959578900250, −1.83957233770709877289524295112, −1.01000571017339276369949623800, 1.60488561404812693188391975590, 2.50588892202341957149153976129, 3.65158592882777264273271355498, 4.77484478179335369302540123919, 5.46332733702856995997139578999, 5.84598974630436377765673290252, 7.03166766925932660887166896231, 8.215814195159878537687215181709, 8.945979307412727425448825149928, 9.427441147989221749465342198191

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