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Properties

Label 2-1792-28.27-c1-0-49
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  + 2.44·3-s − 2.44i·5-s + (2.44 − i)7-s + 2.99·9-s − 2i·11-s − 2.44i·13-s − 5.99i·15-s + 4.89i·17-s − 2.44·19-s + (5.99 − 2.44i)21-s − 4i·23-s − 0.999·25-s + 4·29-s + 4.89·31-s − 4.89i·33-s + ⋯
L(s)  = 1  + 1.41·3-s − 1.09i·5-s + (0.925 − 0.377i)7-s + 0.999·9-s − 0.603i·11-s − 0.679i·13-s − 1.54i·15-s + 1.18i·17-s − 0.561·19-s + (1.30 − 0.534i)21-s − 0.834i·23-s − 0.199·25-s + 0.742·29-s + 0.879·31-s − 0.852i·33-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

\(L(1)\) \(\approx\) \(3.084632263\)
\(L(\frac12)\) \(\approx\) \(3.084632263\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.44T + 3T^{2} \)
5 \( 1 + 2.44iT - 5T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 + 2.44T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 4.89T + 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 4.89T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 - 2.44T + 59T^{2} \)
61 \( 1 + 7.34iT - 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 - 10iT - 71T^{2} \)
73 \( 1 - 14.6iT - 73T^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 - 2.44T + 83T^{2} \)
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

−8.625900788160267614341615553117, −8.422810686289413578286156060068, −8.116554340662671246009462246981, −6.95530542648761686481935987406, −5.81347232657177139814617120271, −4.80363124088719428589376227549, −4.12277068882847408310164497820, −3.15454358769357933207399878380, −2.04039073266894559225728347601, −1.01050733886054927448279798545, 1.79214795135285516117430169771, 2.51231382936894167438888931944, 3.28809623886295478360444267694, 4.32090201298362643699747114542, 5.21434628590146253874550988409, 6.55745040968756918882947676452, 7.22217983030891246768656670719, 7.85151292590505657359425979366, 8.691309373609650826825604557208, 9.244372137528624851637911493154

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