↑

Properties

Label 2-402-201.101-c1-0-5
Degree $2$
Conductor $402$
Sign $-0.845 - 0.534i$
Analytic cond. $3.20998$
Root an. cond. $1.79164$
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  + (0.0475 + 0.998i)2-s + (−1.73 − 0.0162i)3-s + (−0.995 + 0.0950i)4-s + (2.86 + 3.31i)5-s + (−0.0661 − 1.73i)6-s + (−0.641 + 1.24i)7-s + (−0.142 − 0.989i)8-s + (2.99 + 0.0564i)9-s + (−3.17 + 3.02i)10-s + (0.0561 − 0.162i)11-s + (1.72 − 0.148i)12-s + (0.497 − 0.632i)13-s + (−1.27 − 0.581i)14-s + (−4.91 − 5.78i)15-s + (0.981 − 0.189i)16-s + (−0.113 + 1.18i)17-s + ⋯
L(s)  = 1  + (0.0336 + 0.706i)2-s + (−0.999 − 0.00940i)3-s + (−0.497 + 0.0475i)4-s + (1.28 + 1.48i)5-s + (−0.0270 − 0.706i)6-s + (−0.242 + 0.470i)7-s + (−0.0503 − 0.349i)8-s + (0.999 + 0.0188i)9-s + (−1.00 + 0.955i)10-s + (0.0169 − 0.0489i)11-s + (0.498 − 0.0428i)12-s + (0.137 − 0.175i)13-s + (−0.340 − 0.155i)14-s + (−1.26 − 1.49i)15-s + (0.245 − 0.0473i)16-s + (−0.0275 + 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(402\)    =    \(2 \cdot 3 \cdot 67\)
Sign: $-0.845 - 0.534i$
Analytic conductor: \(3.20998\)
Root analytic conductor: \(1.79164\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{402} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 402,\ (\ :1/2),\ -0.845 - 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.291735 + 1.00801i\)
\(L(\frac12)\) \(\approx\) \(0.291735 + 1.00801i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0475 - 0.998i)T \)
3 \( 1 + (1.73 + 0.0162i)T \)
67 \( 1 + (-7.19 - 3.91i)T \)
good5 \( 1 + (-2.86 - 3.31i)T + (-0.711 + 4.94i)T^{2} \)
7 \( 1 + (0.641 - 1.24i)T + (-4.06 - 5.70i)T^{2} \)
11 \( 1 + (-0.0561 + 0.162i)T + (-8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.497 + 0.632i)T + (-3.06 - 12.6i)T^{2} \)
17 \( 1 + (0.113 - 1.18i)T + (-16.6 - 3.21i)T^{2} \)
19 \( 1 + (4.37 - 2.25i)T + (11.0 - 15.4i)T^{2} \)
23 \( 1 + (-3.32 - 0.806i)T + (20.4 + 10.5i)T^{2} \)
29 \( 1 + (7.95 - 4.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.66 + 4.65i)T + (-7.30 + 30.1i)T^{2} \)
37 \( 1 + (-5.20 + 9.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.16 - 3.04i)T + (-13.4 - 38.7i)T^{2} \)
43 \( 1 + (-3.47 + 1.58i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (1.95 - 2.04i)T + (-2.23 - 46.9i)T^{2} \)
53 \( 1 + (-1.94 + 4.26i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (-7.66 + 1.10i)T + (56.6 - 16.6i)T^{2} \)
61 \( 1 + (-9.80 + 3.39i)T + (47.9 - 37.7i)T^{2} \)
71 \( 1 + (-1.15 - 12.0i)T + (-69.7 + 13.4i)T^{2} \)
73 \( 1 + (3.80 + 10.9i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (-2.07 - 5.17i)T + (-57.1 + 54.5i)T^{2} \)
83 \( 1 + (2.73 + 14.2i)T + (-77.0 + 30.8i)T^{2} \)
89 \( 1 + (-0.900 + 3.06i)T + (-74.8 - 48.1i)T^{2} \)
97 \( 1 + (-5.34 - 3.08i)T + (48.5 + 84.0i)T^{2} \)
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

−11.32285939161048397681421916549, −10.73823012376976230206585002178, −9.886160531500991613015552910795, −9.112605336830163027615413585300, −7.48484436192408939092159689666, −6.71893162335210696885010422937, −5.91977759944499212719686662623, −5.47556065398157410588195902517, −3.74093281561444783380067107085, −2.10912704322374312657574213659, 0.78751081680464774927159975375, 2.02106458327443329071298319428, 4.16278347987949934352008835457, 5.00663373373970813544922826750, 5.80567924778143358583892305310, 6.86842203002150757882584819454, 8.483790617146334310760068014413, 9.389965635133871645580323458753, 9.994237265067165801664906572656, 10.88795607415254174228186683466

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