↑

Properties

Label 2-1352-104.43-c0-0-4
Degree $2$
Conductor $1352$
Sign $-0.596 + 0.802i$
Analytic cond. $0.674735$
Root an. cond. $0.821423$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.623 + 1.07i)3-s + (0.499 + 0.866i)4-s + (1.07 − 0.623i)6-s − 0.999i·8-s + (−0.277 − 0.480i)9-s + (−1.56 − 0.900i)11-s − 1.24·12-s + (−0.5 + 0.866i)16-s + (−0.900 − 1.56i)17-s + 0.554i·18-s + (−0.385 + 0.222i)19-s + (0.900 + 1.56i)22-s + (1.07 + 0.623i)24-s − 25-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.623 + 1.07i)3-s + (0.499 + 0.866i)4-s + (1.07 − 0.623i)6-s − 0.999i·8-s + (−0.277 − 0.480i)9-s + (−1.56 − 0.900i)11-s − 1.24·12-s + (−0.5 + 0.866i)16-s + (−0.900 − 1.56i)17-s + 0.554i·18-s + (−0.385 + 0.222i)19-s + (0.900 + 1.56i)22-s + (1.07 + 0.623i)24-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-0.596 + 0.802i$
Analytic conductor: \(0.674735\)
Root analytic conductor: \(0.821423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :0),\ -0.596 + 0.802i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1421718349\)
\(L(\frac12)\) \(\approx\) \(0.1421718349\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 \)
good3 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.56 + 0.900i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.385 - 0.222i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (1.07 - 0.623i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.07 + 0.623i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + 1.24iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 0.445iT - T^{2} \)
89 \( 1 + (-1.07 - 0.623i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.56 - 0.900i)T + (0.5 - 0.866i)T^{2} \)
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

−9.632718445454754541378819914945, −8.971570414344572560026463410336, −8.041665161748697317445237049346, −7.33519937975555212919065800407, −6.13796972748253323907070005654, −5.22382069402436892448897180431, −4.36301955053214044698353367090, −3.25569978112661626189585210587, −2.29229608417114730449157757588, −0.15916503390805092625711964738, 1.62906300060961660240996155430, 2.42865233572076098843251811570, 4.38736912306102406526292517162, 5.51460501894954025847788598481, 6.16891234422377579305978463478, 6.91404231772175221759519533553, 7.68771948597329534220631184378, 8.152846650799784578551943844993, 9.182343877066165669523147633932, 10.15553142980522196920979369332

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