Properties

Label 2-1792-1792.69-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 + (−0.653 − 0.163i)11-s + (0.980 − 0.195i)14-s + (−0.923 − 0.382i)16-s + (−0.923 + 0.382i)18-s + (0.288 + 0.609i)22-s + (1.21 − 1.47i)23-s + (0.0980 − 0.995i)25-s + (−0.773 − 0.634i)28-s + (1.15 + 0.690i)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 + (−0.653 − 0.163i)11-s + (0.980 − 0.195i)14-s + (−0.923 − 0.382i)16-s + (−0.923 + 0.382i)18-s + (0.288 + 0.609i)22-s + (1.21 − 1.47i)23-s + (0.0980 − 0.995i)25-s + (−0.773 − 0.634i)28-s + (1.15 + 0.690i)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: $\chi_{1792} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.492 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7616472374\)
\(L(\frac12)\) \(\approx\) \(0.7616472374\)
\(L(1)\) 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 \( 1 + (0.653 + 0.163i)T + (0.881 + 0.471i)T^{2} \)
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 \( 1 + (-1.15 - 0.690i)T + (0.471 + 0.881i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (-1.75 - 0.829i)T + (0.634 + 0.773i)T^{2} \)
41 \( 1 + (0.980 - 0.195i)T^{2} \)
43 \( 1 + (0.612 - 0.825i)T + (-0.290 - 0.956i)T^{2} \)
47 \( 1 + (0.382 + 0.923i)T^{2} \)
53 \( 1 + (-1.69 + 1.01i)T + (0.471 - 0.881i)T^{2} \)
59 \( 1 + (-0.995 - 0.0980i)T^{2} \)
61 \( 1 + (0.956 + 0.290i)T^{2} \)
67 \( 1 + (-1.17 - 0.174i)T + (0.956 + 0.290i)T^{2} \)
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

−9.462677614705734282660913475640, −8.531626358742714491591450322160, −8.255937761022239278427820336075, −6.86815088678699582766197267502, −6.43448188792829630307335823714, −5.12092961345361648561624065923, −4.19446553931080649066099538664, −3.00718445629564330663949454313, −2.53193669298064534895852093114, −0.885349080454054491811738714007, 1.14843602545972432043440553591, 2.53688615850080257999862855732, 3.95362970917015036715353142551, 4.94379673201982086558369984363, 5.58894392323883486520427724892, 6.66956240116599309209156681409, 7.45655958323899831770237522948, 7.72178176085761057213366214793, 8.772410840881450768726766102725, 9.628686636426351816324732191754

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