Properties

Label 2-322-23.2-c1-0-5
Degree $2$
Conductor $322$
Sign $-0.159 - 0.987i$
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 + (0.341 + 2.37i)3-s + (0.415 + 0.909i)4-s + (1.96 + 0.577i)5-s + (−0.995 + 2.18i)6-s + (0.654 − 0.755i)7-s + (−0.142 + 0.989i)8-s + (−2.63 + 0.774i)9-s + (1.34 + 1.54i)10-s + (0.712 − 0.457i)11-s + (−2.01 + 1.29i)12-s + (−3.16 − 3.65i)13-s + (0.959 − 0.281i)14-s + (−0.699 + 4.86i)15-s + (−0.654 + 0.755i)16-s + (1.83 − 4.02i)17-s + ⋯
L(s)  = 1  + (0.594 + 0.382i)2-s + (0.196 + 1.37i)3-s + (0.207 + 0.454i)4-s + (0.879 + 0.258i)5-s + (−0.406 + 0.890i)6-s + (0.247 − 0.285i)7-s + (−0.0503 + 0.349i)8-s + (−0.878 + 0.258i)9-s + (0.424 + 0.489i)10-s + (0.214 − 0.138i)11-s + (−0.582 + 0.374i)12-s + (−0.878 − 1.01i)13-s + (0.256 − 0.0752i)14-s + (−0.180 + 1.25i)15-s + (−0.163 + 0.188i)16-s + (0.445 − 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.159 - 0.987i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.159 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38868 + 1.63081i\)
\(L(\frac12)\) \(\approx\) \(1.38868 + 1.63081i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.09 - 2.49i)T \)
good3 \( 1 + (-0.341 - 2.37i)T + (-2.87 + 0.845i)T^{2} \)
5 \( 1 + (-1.96 - 0.577i)T + (4.20 + 2.70i)T^{2} \)
11 \( 1 + (-0.712 + 0.457i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (3.16 + 3.65i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (-1.83 + 4.02i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (1.18 + 2.60i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (1.55 - 3.41i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-0.210 + 1.46i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (0.504 - 0.148i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (4.19 + 1.23i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-1.11 - 7.76i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 - 6.67T + 47T^{2} \)
53 \( 1 + (-7.74 + 8.93i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (5.02 + 5.79i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (0.355 - 2.47i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (-9.35 - 6.01i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (-8.70 - 5.59i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (4.45 + 9.74i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (-4.16 - 4.80i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (9.69 - 2.84i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-1.59 - 11.1i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (9.05 + 2.65i)T + (81.6 + 52.4i)T^{2} \)
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

−11.81268225044924107425842877344, −10.77252415902026766031829708885, −9.958370367989291920496031939427, −9.374950172242399048011381232612, −8.080734418493323165759684070074, −6.94805287757240998492302974743, −5.59648781694055445327206141287, −4.96258359393073872883222844269, −3.78323184494815790433349674148, −2.62571945009917236193668784465, 1.65685525174114163038226332497, 2.28553630273922040390813422699, 4.16396813999376847516994932080, 5.60597455070433967254727821320, 6.36079421027763368606490184142, 7.36778762997482444199997647934, 8.458287723360067226861250637096, 9.570221266660872843125153179781, 10.49827462444817026582760703812, 11.98923528008960163606319546598

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