↑

Properties

Label 2-406-7.2-c1-0-9
Degree $2$
Conductor $406$
Sign $0.153 + 0.988i$
Analytic cond. $3.24192$
Root an. cond. $1.80053$
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  + (−0.5 + 0.866i)2-s + (−0.257 − 0.446i)3-s + (−0.499 − 0.866i)4-s + (−1.84 + 3.19i)5-s + 0.515·6-s + (−2.63 + 0.239i)7-s + 0.999·8-s + (1.36 − 2.36i)9-s + (−1.84 − 3.19i)10-s + (−3.22 − 5.58i)11-s + (−0.257 + 0.446i)12-s + 2.84·13-s + (1.10 − 2.40i)14-s + 1.89·15-s + (−0.5 + 0.866i)16-s + (0.767 + 1.32i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.148 − 0.257i)3-s + (−0.249 − 0.433i)4-s + (−0.824 + 1.42i)5-s + 0.210·6-s + (−0.995 + 0.0907i)7-s + 0.353·8-s + (0.455 − 0.789i)9-s + (−0.582 − 1.00i)10-s + (−0.972 − 1.68i)11-s + (−0.0743 + 0.128i)12-s + 0.790·13-s + (0.296 − 0.641i)14-s + 0.490·15-s + (−0.125 + 0.216i)16-s + (0.186 + 0.322i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(406\)    =    \(2 \cdot 7 \cdot 29\)
Sign: $0.153 + 0.988i$
Analytic conductor: \(3.24192\)
Root analytic conductor: \(1.80053\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{406} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 406,\ (\ :1/2),\ 0.153 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.296043 - 0.253582i\)
\(L(\frac12)\) \(\approx\) \(0.296043 - 0.253582i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.63 - 0.239i)T \)
29 \( 1 + T \)
good3 \( 1 + (0.257 + 0.446i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.84 - 3.19i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.22 + 5.58i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.84T + 13T^{2} \)
17 \( 1 + (-0.767 - 1.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.95 + 3.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.190 - 0.329i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (3.99 + 6.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.73 + 6.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.98T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + (3.92 - 6.80i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.610 - 1.05i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.348 - 0.603i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.55 + 2.68i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.61 - 6.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + (7.47 + 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.56 - 4.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + (4.69 - 8.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.56T + 97T^{2} \)
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

−11.01290639471973612712293849619, −10.20016723598388347801482221058, −9.144668428068281119888153407478, −8.065226884468247148355438842321, −7.23529810268669528134317630820, −6.39424294498182154734559114825, −5.79935677165732139691064893979, −3.74158905185707449796251631970, −3.04848206900724805586044204226, −0.29360496097624797365963379682, 1.63605964822940966429993954603, 3.45721348076305865215755778315, 4.56359840890855882429568542170, 5.24772587712133580080574865516, 7.10200744175920989016878536594, 7.939308745588966091329397574552, 8.779089606762512885569514370948, 9.938202570983712947624196300820, 10.20973099205797420300191776150, 11.57823400309599111188091543261

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