Properties

Label 2-1792-28.27-c1-0-19
Degree $2$
Conductor $1792$
Sign $0.755 - 0.654i$
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  − 0.732·3-s + 2.73i·5-s + (2 − 1.73i)7-s − 2.46·9-s + 1.46i·11-s − 1.26i·13-s − 2i·15-s − 4i·17-s + 4.73·19-s + (−1.46 + 1.26i)21-s − 1.46i·23-s − 2.46·25-s + 4·27-s + 6.92·29-s + 6.92·31-s + ⋯
L(s)  = 1  − 0.422·3-s + 1.22i·5-s + (0.755 − 0.654i)7-s − 0.821·9-s + 0.441i·11-s − 0.351i·13-s − 0.516i·15-s − 0.970i·17-s + 1.08·19-s + (−0.319 + 0.276i)21-s − 0.305i·23-s − 0.492·25-s + 0.769·27-s + 1.28·29-s + 1.24·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ 0.755 - 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.517166138\)
\(L(\frac12)\) \(\approx\) \(1.517166138\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.732T + 3T^{2} \)
5 \( 1 - 2.73iT - 5T^{2} \)
11 \( 1 - 1.46iT - 11T^{2} \)
13 \( 1 + 1.26iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 4.73T + 19T^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 10.9iT - 41T^{2} \)
43 \( 1 - 9.46iT - 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 6.92T + 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 + 5.66iT - 61T^{2} \)
67 \( 1 - 9.46iT - 67T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 - 7.26T + 83T^{2} \)
89 \( 1 - 1.07iT - 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−9.634611055441581144336654905634, −8.331296026421885016985619968383, −7.79424689992802731710705067168, −6.85165027903670139252040360106, −6.39817283727256316719675304594, −5.19964954310748762484086287959, −4.62132978223779372232419129059, −3.22002225352805589147018759302, −2.64157986983474498769625291627, −0.996521312305338989560089000490, 0.78831150340855633856001371166, 1.93689706775363467564271433827, 3.24895834688436860532750468647, 4.47728641549432196620877133598, 5.25055608999607256001843169193, 5.68269782999461804617552959280, 6.65127397876537475189678540714, 7.902367443474173851548421253099, 8.635563529335969910465552003616, 8.797386924713586596707751590123

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