Properties

Label 2-1792-8.5-c1-0-41
Degree $2$
Conductor $1792$
Sign $-0.707 + 0.707i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2i·5-s − 7-s + 3·9-s − 4i·11-s + 2i·13-s − 6·17-s − 8i·19-s + 25-s + 6i·29-s − 8·31-s + 2i·35-s + 2i·37-s − 2·41-s − 4i·43-s − 6i·45-s + ⋯
L(s)  = 1  − 0.894i·5-s − 0.377·7-s + 9-s − 1.20i·11-s + 0.554i·13-s − 1.45·17-s − 1.83i·19-s + 0.200·25-s + 1.11i·29-s − 1.43·31-s + 0.338i·35-s + 0.328i·37-s − 0.312·41-s − 0.609i·43-s − 0.894i·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.148105728\)
\(L(\frac12)\) \(\approx\) \(1.148105728\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 - 3T^{2} \)
5 \( 1 + 2iT - 5T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 6iT - 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.890938324018158262513287189505, −8.582266227698537095806042349993, −7.11196304633871858201230720088, −6.84466491763944435938810317878, −5.67284219814268850127895572559, −4.78320428605617232983467798508, −4.13262069038847401481089269234, −2.98388876822502447763400489625, −1.69115921376259029597674795708, −0.41698006543836759305244622919, 1.67703211091000580964232125707, 2.61355728361102460213543875210, 3.83905909196373736255552550968, 4.42345508652110506433275110732, 5.68784206126345907746742174977, 6.50588879532471922078838354857, 7.25214299997128106071549019140, 7.70571146092749935795184798176, 8.903712408776508411217315203105, 9.745836656439833316282704051939

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