Properties

Label 2-1792-1792.13-c0-0-0
Degree $2$
Conductor $1792$
Sign $-0.492 + 0.870i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.634 − 0.773i)2-s + (−0.195 − 0.980i)4-s + (0.471 + 0.881i)7-s + (−0.881 − 0.471i)8-s + (−0.290 − 0.956i)9-s + (−0.457 − 1.82i)11-s + (0.980 + 0.195i)14-s + (−0.923 + 0.382i)16-s + (−0.923 − 0.382i)18-s + (−1.70 − 0.805i)22-s + (1.21 + 1.47i)23-s + (−0.0980 − 0.995i)25-s + (0.773 − 0.634i)28-s + (−0.761 − 1.27i)29-s + (−0.290 + 0.956i)32-s + ⋯
L(s)  = 1  + (0.634 − 0.773i)2-s + (−0.195 − 0.980i)4-s + (0.471 + 0.881i)7-s + (−0.881 − 0.471i)8-s + (−0.290 − 0.956i)9-s + (−0.457 − 1.82i)11-s + (0.980 + 0.195i)14-s + (−0.923 + 0.382i)16-s + (−0.923 − 0.382i)18-s + (−1.70 − 0.805i)22-s + (1.21 + 1.47i)23-s + (−0.0980 − 0.995i)25-s + (0.773 − 0.634i)28-s + (−0.761 − 1.27i)29-s + (−0.290 + 0.956i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.492 + 0.870i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ -0.492 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.448128270\)
\(L(\frac12)\) \(\approx\) \(1.448128270\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.634 + 0.773i)T \)
7 \( 1 + (-0.471 - 0.881i)T \)
good3 \( 1 + (0.290 + 0.956i)T^{2} \)
5 \( 1 + (0.0980 + 0.995i)T^{2} \)
11 \( 1 + (0.457 + 1.82i)T + (-0.881 + 0.471i)T^{2} \)
13 \( 1 + (0.995 + 0.0980i)T^{2} \)
17 \( 1 + (0.923 + 0.382i)T^{2} \)
19 \( 1 + (-0.773 + 0.634i)T^{2} \)
23 \( 1 + (-1.21 - 1.47i)T + (-0.195 + 0.980i)T^{2} \)
29 \( 1 + (0.761 + 1.27i)T + (-0.471 + 0.881i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.207 - 0.439i)T + (-0.634 + 0.773i)T^{2} \)
41 \( 1 + (0.980 + 0.195i)T^{2} \)
43 \( 1 + (-1.37 + 1.02i)T + (0.290 - 0.956i)T^{2} \)
47 \( 1 + (0.382 - 0.923i)T^{2} \)
53 \( 1 + (-0.150 + 0.251i)T + (-0.471 - 0.881i)T^{2} \)
59 \( 1 + (0.995 - 0.0980i)T^{2} \)
61 \( 1 + (-0.956 + 0.290i)T^{2} \)
67 \( 1 + (-0.235 - 1.58i)T + (-0.956 + 0.290i)T^{2} \)
71 \( 1 + (1.59 + 0.482i)T + (0.831 + 0.555i)T^{2} \)
73 \( 1 + (0.555 + 0.831i)T^{2} \)
79 \( 1 + (-1.65 - 1.10i)T + (0.382 + 0.923i)T^{2} \)
83 \( 1 + (0.634 + 0.773i)T^{2} \)
89 \( 1 + (0.195 + 0.980i)T^{2} \)
97 \( 1 + (0.707 - 0.707i)T^{2} \)
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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

−9.110065835771781513075519681290, −8.781796773734692229321362125287, −7.76386113797882651866578673438, −6.40209201071776559722028997061, −5.75940330777821364304488646180, −5.29117431182318886803585066115, −4.00643999573569765156342270574, −3.19871565423191412931761375667, −2.41839525252736730875433165490, −0.916880922059763149526283814363, 1.93782750958841770822411692477, 3.04156839435188720168257839594, 4.37005622882323528286721770234, 4.76085913244221665318020255317, 5.50427937045206980544132014866, 6.76895682851479338107249088432, 7.38822058167198482927997046729, 7.75664018378767819067155651920, 8.768643395555571938991928483594, 9.595144886302802029482827641303

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