Properties

Degree $2$
Conductor $1792$
Sign $0.707 + 0.707i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.55·3-s + 9.86i·5-s + 2.64i·7-s − 2.47·9-s − 13.1·11-s + 5.86i·13-s − 25.2i·15-s − 0.570·17-s − 15.6·19-s − 6.75i·21-s − 16.4i·23-s − 72.3·25-s + 29.3·27-s + 29.7i·29-s − 54.8i·31-s + ⋯
L(s)  = 1  − 0.851·3-s + 1.97i·5-s + 0.377i·7-s − 0.274·9-s − 1.19·11-s + 0.451i·13-s − 1.68i·15-s − 0.0335·17-s − 0.824·19-s − 0.321i·21-s − 0.716i·23-s − 2.89·25-s + 1.08·27-s + 1.02i·29-s − 1.76i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.707 + 0.707i$
Motivic weight: \(2\)
Character: $\chi_{1792} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1591462991\)
\(L(\frac12)\) \(\approx\) \(0.1591462991\)
\(L(2)\) 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 \( 1 - 2.64iT \)
good3 \( 1 + 2.55T + 9T^{2} \)
5 \( 1 - 9.86iT - 25T^{2} \)
11 \( 1 + 13.1T + 121T^{2} \)
13 \( 1 - 5.86iT - 169T^{2} \)
17 \( 1 + 0.570T + 289T^{2} \)
19 \( 1 + 15.6T + 361T^{2} \)
23 \( 1 + 16.4iT - 529T^{2} \)
29 \( 1 - 29.7iT - 841T^{2} \)
31 \( 1 + 54.8iT - 961T^{2} \)
37 \( 1 + 42.0iT - 1.36e3T^{2} \)
41 \( 1 + 0.773T + 1.68e3T^{2} \)
43 \( 1 + 41.7T + 1.84e3T^{2} \)
47 \( 1 - 58.4iT - 2.20e3T^{2} \)
53 \( 1 - 5.65iT - 2.80e3T^{2} \)
59 \( 1 + 42.6T + 3.48e3T^{2} \)
61 \( 1 - 95.9iT - 3.72e3T^{2} \)
67 \( 1 - 69.8T + 4.48e3T^{2} \)
71 \( 1 - 92.0iT - 5.04e3T^{2} \)
73 \( 1 + 9.97T + 5.32e3T^{2} \)
79 \( 1 + 20.1iT - 6.24e3T^{2} \)
83 \( 1 - 151.T + 6.88e3T^{2} \)
89 \( 1 + 5.79T + 7.92e3T^{2} \)
97 \( 1 - 103.T + 9.40e3T^{2} \)
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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

−9.059622837654926986029771131072, −7.994035456234957080294012934800, −7.29420728635497371603287164728, −6.40725232722158575759606704484, −6.03227861696028648022886884620, −5.10882146202701959117744760525, −3.91674954307099309046137590575, −2.77781002852378292833513458342, −2.27106362247830295978068703043, −0.06935936660550567175887235959, 0.68805362026780745509060067157, 1.86861760978213009302467036206, 3.40579603816687196859250585006, 4.69882532030620100817011758005, 5.05238773457060479527331432169, 5.70795842623471733436094692557, 6.63948621346319581961593610755, 7.993690427572302441621555348980, 8.242771774846597189846476831953, 9.119639924527825943457932361549

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