Properties

Label 2-322-23.12-c1-0-1
Degree $2$
Conductor $322$
Sign $-0.722 - 0.691i$
Analytic cond. $2.57118$
Root an. cond. $1.60349$
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  + (0.841 − 0.540i)2-s + (−0.473 + 3.29i)3-s + (0.415 − 0.909i)4-s + (−2.68 + 0.789i)5-s + (1.38 + 3.02i)6-s + (0.654 + 0.755i)7-s + (−0.142 − 0.989i)8-s + (−7.74 − 2.27i)9-s + (−1.83 + 2.11i)10-s + (2.03 + 1.31i)11-s + (2.79 + 1.79i)12-s + (−1.56 + 1.80i)13-s + (0.959 + 0.281i)14-s + (−1.32 − 9.22i)15-s + (−0.654 − 0.755i)16-s + (−1.04 − 2.28i)17-s + ⋯
L(s)  = 1  + (0.594 − 0.382i)2-s + (−0.273 + 1.90i)3-s + (0.207 − 0.454i)4-s + (−1.20 + 0.352i)5-s + (0.564 + 1.23i)6-s + (0.247 + 0.285i)7-s + (−0.0503 − 0.349i)8-s + (−2.58 − 0.758i)9-s + (−0.579 + 0.669i)10-s + (0.614 + 0.395i)11-s + (0.808 + 0.519i)12-s + (−0.434 + 0.500i)13-s + (0.256 + 0.0752i)14-s + (−0.342 − 2.38i)15-s + (−0.163 − 0.188i)16-s + (−0.252 − 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.722 - 0.691i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.722 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.415096 + 1.03375i\)
\(L(\frac12)\) \(\approx\) \(0.415096 + 1.03375i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.841 + 0.540i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (-4.38 + 1.95i)T \)
good3 \( 1 + (0.473 - 3.29i)T + (-2.87 - 0.845i)T^{2} \)
5 \( 1 + (2.68 - 0.789i)T + (4.20 - 2.70i)T^{2} \)
11 \( 1 + (-2.03 - 1.31i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (1.56 - 1.80i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (1.04 + 2.28i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (2.73 - 5.98i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-2.32 - 5.09i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-0.688 - 4.79i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (2.54 + 0.746i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (-10.2 + 3.01i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (0.540 - 3.75i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 - 4.68T + 47T^{2} \)
53 \( 1 + (-9.27 - 10.7i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-7.89 + 9.11i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (-0.362 - 2.51i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (3.50 - 2.25i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (6.22 - 4.00i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-1.08 + 2.36i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-5.38 + 6.21i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (16.2 + 4.76i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-0.504 + 3.50i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (8.20 - 2.41i)T + (81.6 - 52.4i)T^{2} \)
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

−11.83784701922926436672492728093, −11.00156481083629589128561242569, −10.43385206199337235110618848481, −9.365513252342313715266373725252, −8.555435899319710970495061618176, −7.01027205444679334981430979378, −5.64226905880142801135159909931, −4.52932337475570086839636153990, −4.03922329530302761670472520252, −2.95280061584083501559641372538, 0.68721493067723082604168090615, 2.57807607887822432872376093281, 4.16106046986932816730758674966, 5.49978095615432035725834778708, 6.61818402990498079620838544448, 7.30776339288172900283143035089, 8.070637490696218834711385758834, 8.769187503459515583353734418121, 11.04080617241080366229045647022, 11.58954154621690801659285547080

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