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Properties

Label 2-1792-16.13-c1-0-44
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  + (−0.236 − 0.236i)3-s + (2.98 − 2.98i)5-s − i·7-s − 2.88i·9-s + (2.55 − 2.55i)11-s + (−2.17 − 2.17i)13-s − 1.41·15-s − 0.473·17-s + (5.12 + 5.12i)19-s + (−0.236 + 0.236i)21-s + 0.0840i·23-s − 12.8i·25-s + (−1.39 + 1.39i)27-s + (−1.35 − 1.35i)29-s + 0.196·31-s + ⋯
L(s)  = 1  + (−0.136 − 0.136i)3-s + (1.33 − 1.33i)5-s − 0.377i·7-s − 0.962i·9-s + (0.771 − 0.771i)11-s + (−0.603 − 0.603i)13-s − 0.365·15-s − 0.114·17-s + (1.17 + 1.17i)19-s + (−0.0516 + 0.0516i)21-s + 0.0175i·23-s − 2.56i·25-s + (−0.268 + 0.268i)27-s + (−0.251 − 0.251i)29-s + 0.0352·31-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

\(L(1)\) \(\approx\) \(2.131942786\)
\(L(\frac12)\) \(\approx\) \(2.131942786\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.236 + 0.236i)T + 3iT^{2} \)
5 \( 1 + (-2.98 + 2.98i)T - 5iT^{2} \)
11 \( 1 + (-2.55 + 2.55i)T - 11iT^{2} \)
13 \( 1 + (2.17 + 2.17i)T + 13iT^{2} \)
17 \( 1 + 0.473T + 17T^{2} \)
19 \( 1 + (-5.12 - 5.12i)T + 19iT^{2} \)
23 \( 1 - 0.0840iT - 23T^{2} \)
29 \( 1 + (1.35 + 1.35i)T + 29iT^{2} \)
31 \( 1 - 0.196T + 31T^{2} \)
37 \( 1 + (-1.74 + 1.74i)T - 37iT^{2} \)
41 \( 1 - 5.81iT - 41T^{2} \)
43 \( 1 + (7.30 - 7.30i)T - 43iT^{2} \)
47 \( 1 - 3.52T + 47T^{2} \)
53 \( 1 + (4.88 - 4.88i)T - 53iT^{2} \)
59 \( 1 + (7.51 - 7.51i)T - 59iT^{2} \)
61 \( 1 + (-5.77 - 5.77i)T + 61iT^{2} \)
67 \( 1 + (-9.77 - 9.77i)T + 67iT^{2} \)
71 \( 1 + 8.95iT - 71T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 - 9.51T + 79T^{2} \)
83 \( 1 + (5.85 + 5.85i)T + 83iT^{2} \)
89 \( 1 + 15.3iT - 89T^{2} \)
97 \( 1 - 11.2T + 97T^{2} \)
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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

−9.150155960042033037576876736305, −8.419534478244619705821507511746, −7.51080830289775914946375341855, −6.30358745353270210464480881677, −5.88924050968087601138857670192, −5.10640653326779517137371771641, −4.10342544393704628385150712416, −3.02210188757646840320078214371, −1.52038069127917236080576975727, −0.844169947545949358291580091123, 1.87205278586325951014488532739, 2.38252304340564849450777939113, 3.47062504635391735752235254799, 4.90268053697151370831518592524, 5.38669168130855080985733370471, 6.56582997486575300432382416110, 6.89903208599461047678132046344, 7.73849284926621216840737250912, 9.120204694564322873697609665633, 9.563062269107394868491891834518

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