↑

Properties

Label 2-3696-3696.2309-c0-0-2
Degree $2$
Conductor $3696$
Sign $-0.608 - 0.793i$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
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  + (−0.258 − 0.965i)2-s + (−0.707 + 0.707i)3-s + (−0.866 + 0.499i)4-s + (1.22 + 1.22i)5-s + (0.866 + 0.500i)6-s + i·7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (0.866 − 1.49i)10-s + (−0.707 − 0.707i)11-s + (0.258 − 0.965i)12-s + (−1.36 + 1.36i)13-s + (0.965 − 0.258i)14-s − 1.73·15-s + (0.500 − 0.866i)16-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.707 + 0.707i)3-s + (−0.866 + 0.499i)4-s + (1.22 + 1.22i)5-s + (0.866 + 0.500i)6-s + i·7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (0.866 − 1.49i)10-s + (−0.707 − 0.707i)11-s + (0.258 − 0.965i)12-s + (−1.36 + 1.36i)13-s + (0.965 − 0.258i)14-s − 1.73·15-s + (0.500 − 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3696\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3696} (2309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3696,\ (\ :0),\ -0.608 - 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6396110720\)
\(L(\frac12)\) \(\approx\) \(0.6396110720\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
13 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + 0.517T + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
61 \( 1 + (1 - i)T - iT^{2} \)
67 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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

−9.411229803103963616930696732844, −8.579590292729540405399654008257, −7.52906734774657336330852035404, −6.40801467291051310392914740735, −6.02668914633212287773913483219, −5.07263566587560247843167765999, −4.47853935085568223264515823858, −3.14494430842250431026259526955, −2.64089964713610377550285416089, −1.81215586058816675648343861395, 0.43961106769313574301414431245, 1.41748917484718602733466337069, 2.54159770141875719903353962383, 4.50734270140451345836227925577, 4.90048008726716105939651537028, 5.48811346537719158340527569303, 6.17089580460406076504229926108, 7.03265012516648354202506838576, 7.66946427201374731933293215367, 8.117693835241407024552294494254

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