↑

Properties

Label 2-1075-215.214-c0-0-0
Degree $2$
Conductor $1075$
Sign $-0.447 - 0.894i$
Analytic cond. $0.536494$
Root an. cond. $0.732458$
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  − 4-s − 9-s − 11-s + i·13-s + 16-s + 2i·17-s + i·23-s − 31-s + 36-s − 41-s − i·43-s + 44-s − i·47-s − 49-s − i·52-s + i·53-s + ⋯
L(s)  = 1  − 4-s − 9-s − 11-s + i·13-s + 16-s + 2i·17-s + i·23-s − 31-s + 36-s − 41-s − i·43-s + 44-s − i·47-s − 49-s − i·52-s + i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1075\)    =    \(5^{2} \cdot 43\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(0.536494\)
Root analytic conductor: \(0.732458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1075} (1074, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1075,\ (\ :0),\ -0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4405458820\)
\(L(\frac12)\) \(\approx\) \(0.4405458820\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
43 \( 1 + iT \)
good2 \( 1 + T^{2} \)
3 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 - 2iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T + T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 2iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT - T^{2} \)
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

−10.29586861878426214020885140214, −9.468979129857954632981059523809, −8.545533292424225122099746665542, −8.239845215912560371337269468684, −7.07670282114318100472183995660, −5.82587646479287050226969921810, −5.34156082856494342582183294563, −4.16508532795359051787973740156, −3.36378360015214391545361399414, −1.86588882259031469186360707591, 0.40089628601487793675720536556, 2.64161187835360875983280018827, 3.41307307487064382764512112776, 5.02968330861519447343451446807, 5.11054448492961609752338549108, 6.30775246697283573519867795762, 7.62473015802855858078951037541, 8.161805663463986362221589331303, 9.020094354777734552898552304458, 9.716501285657356143073123786634

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