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Properties

Label 2-1792-16.13-c1-0-1
Degree $2$
Conductor $1792$
Sign $-0.382 - 0.923i$
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  + (−0.898 − 0.898i)3-s + (0.786 − 0.786i)5-s + i·7-s − 1.38i·9-s + (−2.15 + 2.15i)11-s + (−1.31 − 1.31i)13-s − 1.41·15-s − 1.79·17-s + (−0.531 − 0.531i)19-s + (0.898 − 0.898i)21-s + 5.95i·23-s + 3.76i·25-s + (−3.94 + 3.94i)27-s + (−5.62 − 5.62i)29-s + 4.34·31-s + ⋯
L(s)  = 1  + (−0.519 − 0.519i)3-s + (0.351 − 0.351i)5-s + 0.377i·7-s − 0.461i·9-s + (−0.650 + 0.650i)11-s + (−0.364 − 0.364i)13-s − 0.365·15-s − 0.436·17-s + (−0.121 − 0.121i)19-s + (0.196 − 0.196i)21-s + 1.24i·23-s + 0.752i·25-s + (−0.758 + 0.758i)27-s + (−1.04 − 1.04i)29-s + 0.779·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1345, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3727384071\)
\(L(\frac12)\) \(\approx\) \(0.3727384071\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 + (0.898 + 0.898i)T + 3iT^{2} \)
5 \( 1 + (-0.786 + 0.786i)T - 5iT^{2} \)
11 \( 1 + (2.15 - 2.15i)T - 11iT^{2} \)
13 \( 1 + (1.31 + 1.31i)T + 13iT^{2} \)
17 \( 1 + 1.79T + 17T^{2} \)
19 \( 1 + (0.531 + 0.531i)T + 19iT^{2} \)
23 \( 1 - 5.95iT - 23T^{2} \)
29 \( 1 + (5.62 + 5.62i)T + 29iT^{2} \)
31 \( 1 - 4.34T + 31T^{2} \)
37 \( 1 + (2.12 - 2.12i)T - 37iT^{2} \)
41 \( 1 - 0.712iT - 41T^{2} \)
43 \( 1 + (2.96 - 2.96i)T - 43iT^{2} \)
47 \( 1 + 2.20T + 47T^{2} \)
53 \( 1 + (-3.38 + 3.38i)T - 53iT^{2} \)
59 \( 1 + (2.41 - 2.41i)T - 59iT^{2} \)
61 \( 1 + (7.09 + 7.09i)T + 61iT^{2} \)
67 \( 1 + (-6.76 - 6.76i)T + 67iT^{2} \)
71 \( 1 - 1.96iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + (5.95 + 5.95i)T + 83iT^{2} \)
89 \( 1 - 16.9iT - 89T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−9.607159697436320053483469599838, −8.802025205847102799910885100725, −7.79715257282842307156794926418, −7.17011941094087745184036480675, −6.25560960313831640848957936214, −5.53000131422166637609800240030, −4.87102938378772231999959597886, −3.64042876588065855614341601996, −2.42972181836609897740625077363, −1.38109620926212028316898158016, 0.14456743821284355324239685537, 1.99939378087703272285589497352, 2.99266534025468567870587983937, 4.22479237345333491891963807425, 4.91142449790652749729234603515, 5.76887969467790972932579228082, 6.54182984991062586527507018165, 7.41050754264797852583179358971, 8.288757752741275147154733407852, 9.062798305674141892907435190337

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