Properties

Label 2-1792-16.13-c1-0-4
Degree $2$
Conductor $1792$
Sign $0.608 - 0.793i$
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  + (−1.48 − 1.48i)3-s + (−1.83 + 1.83i)5-s i·7-s + 1.40i·9-s + (−0.321 + 0.321i)11-s + (−4.61 − 4.61i)13-s + 5.45·15-s − 1.84·17-s + (3.88 + 3.88i)19-s + (−1.48 + 1.48i)21-s − 5.88i·23-s − 1.74i·25-s + (−2.36 + 2.36i)27-s + (−6.14 − 6.14i)29-s + 5.69·31-s + ⋯
L(s)  = 1  + (−0.857 − 0.857i)3-s + (−0.821 + 0.821i)5-s − 0.377i·7-s + 0.469i·9-s + (−0.0969 + 0.0969i)11-s + (−1.28 − 1.28i)13-s + 1.40·15-s − 0.446·17-s + (0.892 + 0.892i)19-s + (−0.323 + 0.323i)21-s − 1.22i·23-s − 0.348i·25-s + (−0.454 + 0.454i)27-s + (−1.14 − 1.14i)29-s + 1.02·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.608 - 0.793i$
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: \((2,\ 1792,\ (\ :1/2),\ 0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4587332367\)
\(L(\frac12)\) \(\approx\) \(0.4587332367\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 + 1.48i)T + 3iT^{2} \)
5 \( 1 + (1.83 - 1.83i)T - 5iT^{2} \)
11 \( 1 + (0.321 - 0.321i)T - 11iT^{2} \)
13 \( 1 + (4.61 + 4.61i)T + 13iT^{2} \)
17 \( 1 + 1.84T + 17T^{2} \)
19 \( 1 + (-3.88 - 3.88i)T + 19iT^{2} \)
23 \( 1 + 5.88iT - 23T^{2} \)
29 \( 1 + (6.14 + 6.14i)T + 29iT^{2} \)
31 \( 1 - 5.69T + 31T^{2} \)
37 \( 1 + (1.66 - 1.66i)T - 37iT^{2} \)
41 \( 1 - 10.7iT - 41T^{2} \)
43 \( 1 + (-0.533 + 0.533i)T - 43iT^{2} \)
47 \( 1 + 0.465T + 47T^{2} \)
53 \( 1 + (-0.623 + 0.623i)T - 53iT^{2} \)
59 \( 1 + (7.32 - 7.32i)T - 59iT^{2} \)
61 \( 1 + (-7.57 - 7.57i)T + 61iT^{2} \)
67 \( 1 + (-6.16 - 6.16i)T + 67iT^{2} \)
71 \( 1 - 0.162iT - 71T^{2} \)
73 \( 1 + 3.49iT - 73T^{2} \)
79 \( 1 - 8.28T + 79T^{2} \)
83 \( 1 + (-2.51 - 2.51i)T + 83iT^{2} \)
89 \( 1 + 1.60iT - 89T^{2} \)
97 \( 1 + 8.88T + 97T^{2} \)
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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

−9.646287076596792723887278970974, −8.158334701508593622749404738399, −7.65279151533458253745236258434, −7.09282838028006447062804813430, −6.32710455040279855204844248000, −5.51205043110657246801123265663, −4.51690290744513882085210126800, −3.40435514734831102401869136955, −2.45407830751629773477077446331, −0.847837850057543543488127176842, 0.26187214099804703937375432162, 2.02611971852467022699657541854, 3.50928268458978458760260554184, 4.46466877821478361923175893999, 4.98416668978056082371077703780, 5.56856269304609387700863156100, 6.85563566017781690984880030244, 7.51312832836153874001307711173, 8.531672490087265844903767754435, 9.409158442878763446933264754545

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