↑

Properties

Label 2-1352-104.3-c0-0-2
Degree $2$
Conductor $1352$
Sign $-0.488 - 0.872i$
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 + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + i·5-s + (−0.866 − 0.499i)6-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−0.5 − 0.866i)10-s + 0.999·12-s − 0.999·14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.499i)20-s + 0.999i·21-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + i·5-s + (−0.866 − 0.499i)6-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−0.5 − 0.866i)10-s + 0.999·12-s − 0.999·14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.499i)20-s + 0.999i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-0.488 - 0.872i$
Analytic conductor: \(0.674735\)
Root analytic conductor: \(0.821423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :0),\ -0.488 - 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9443587830\)
\(L(\frac12)\) \(\approx\) \(0.9443587830\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - iT - T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + 2iT - T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−10.00761288000489075628467644345, −9.216575187813700785456121003875, −8.500240611808552294907969611522, −7.87228676368654740599563913634, −6.86394049870925059640373243375, −6.15466615951639683696845905163, −5.11764075521629302976500831717, −4.09573525648266085899793284969, −2.89077983476678730281194608436, −1.85795936069886084735672758717, 1.10937426493994977966698199142, 1.85365870457122038263774528648, 3.01631463750660247359219123394, 4.37117090489115931748661513774, 5.15861076468363734090969254642, 6.78268335197455274205770368909, 7.30868573119651925298810960927, 8.118582216476230044149449340464, 8.655813226829950524281851414126, 9.251494028181193775628189782953

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