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Properties

Label 2-1792-16.5-c1-0-14
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: $\chi_{1792} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.702826889\)
\(L(\frac12)\) \(\approx\) \(1.702826889\)
\(L(\frac{3}{2})\) 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

−9.749243180611033266475174648880, −8.746541732907067628584468305243, −7.79962632079975296867705668986, −7.16191534490667816179820266433, −6.26253455579829456628368314828, −5.72106832290701330572800419660, −4.70140047295233617013438514670, −3.40360842271398285371773633796, −2.44464245034341192700562490956, −1.84490795149228609450499035130, 0.56452228844791564478928001879, 1.99452108800321007239102896319, 2.72215824581238446567661253217, 4.32908631210320721115835695786, 4.95347753620340839050079538973, 5.72481472799542738664548686259, 6.44664206143573987731148258553, 7.50517804954892538433869833590, 8.645095979695441771376127767143, 8.949696348259161238652160413504

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