Properties

Label 2-1792-28.27-c1-0-7
Degree $2$
Conductor $1792$
Sign $0.377 - 0.925i$
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  − 1.41·3-s − 1.41i·5-s + (2.44 + i)7-s − 0.999·9-s − 3.46i·11-s + 4.24i·13-s + 2.00i·15-s + 4.89i·17-s − 4.24·19-s + (−3.46 − 1.41i)21-s + 2.99·25-s + 5.65·27-s − 4.89·31-s + 4.89i·33-s + (1.41 − 3.46i)35-s + ⋯
L(s)  = 1  − 0.816·3-s − 0.632i·5-s + (0.925 + 0.377i)7-s − 0.333·9-s − 1.04i·11-s + 1.17i·13-s + 0.516i·15-s + 1.18i·17-s − 0.973·19-s + (−0.755 − 0.308i)21-s + 0.599·25-s + 1.08·27-s − 0.879·31-s + 0.852i·33-s + (0.239 − 0.585i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.377 - 0.925i$
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: \((2,\ 1792,\ (\ :1/2),\ 0.377 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9901323085\)
\(L(\frac12)\) \(\approx\) \(0.9901323085\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.44 - i)T \)
good3 \( 1 + 1.41T + 3T^{2} \)
5 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4.89T + 31T^{2} \)
37 \( 1 + 6.92T + 37T^{2} \)
41 \( 1 - 9.79iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 + 7.07T + 59T^{2} \)
61 \( 1 + 4.24iT - 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 + 14iT - 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 - 14.6iT - 89T^{2} \)
97 \( 1 - 4.89iT - 97T^{2} \)
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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

−9.101231394121912519320832628374, −8.670866224922697011030839522669, −8.124524959175274353183681384144, −6.84927244468850244783896315876, −6.07143705023764871286833757130, −5.42439350223141588381398451816, −4.65920918235733389131971794140, −3.76285928726113166793751585330, −2.25470791701218558743337775310, −1.14373472265788784574035500827, 0.46051135404297379575111914218, 2.03092907925737576382908875225, 3.09311379520364542955080406735, 4.35422755938725261962571537028, 5.15031428424166820924644828550, 5.74877141663489517745530987038, 6.95454134128829028468864332001, 7.28287612589443794313176036463, 8.287100134043885480362392406568, 9.106959866151693886559920976930

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