Properties

Label 2-322-161.10-c1-0-9
Degree $2$
Conductor $322$
Sign $0.998 - 0.0538i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.327 − 0.945i)2-s + (1.26 + 2.45i)3-s + (−0.786 − 0.618i)4-s + (2.91 − 0.278i)5-s + (2.73 − 0.392i)6-s + (1.42 − 2.22i)7-s + (−0.841 + 0.540i)8-s + (−2.67 + 3.75i)9-s + (0.691 − 2.84i)10-s + (−1.73 + 0.601i)11-s + (0.522 − 2.70i)12-s + (−0.0383 − 0.130i)13-s + (−1.63 − 2.07i)14-s + (4.37 + 6.80i)15-s + (0.235 + 0.971i)16-s + (−6.11 − 2.44i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (0.729 + 1.41i)3-s + (−0.393 − 0.309i)4-s + (1.30 − 0.124i)5-s + (1.11 − 0.160i)6-s + (0.539 − 0.841i)7-s + (−0.297 + 0.191i)8-s + (−0.891 + 1.25i)9-s + (0.218 − 0.900i)10-s + (−0.524 + 0.181i)11-s + (0.150 − 0.781i)12-s + (−0.0106 − 0.0362i)13-s + (−0.437 − 0.555i)14-s + (1.12 + 1.75i)15-s + (0.0589 + 0.242i)16-s + (−1.48 − 0.593i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.998 - 0.0538i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.998 - 0.0538i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07661 + 0.0559493i\)
\(L(\frac12)\) \(\approx\) \(2.07661 + 0.0559493i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.327 + 0.945i)T \)
7 \( 1 + (-1.42 + 2.22i)T \)
23 \( 1 + (-4.60 + 1.33i)T \)
good3 \( 1 + (-1.26 - 2.45i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (-2.91 + 0.278i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (1.73 - 0.601i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (0.0383 + 0.130i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (6.11 + 2.44i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (3.14 - 1.26i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-1.04 - 7.27i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-5.00 + 0.238i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (3.30 + 2.35i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (2.24 + 1.02i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (3.81 - 5.94i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (0.799 - 0.461i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.61 + 10.0i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (0.902 + 0.218i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (11.5 + 5.95i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-0.168 - 0.875i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-7.84 - 9.05i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-1.32 + 1.68i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-10.7 + 11.3i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-5.36 - 11.7i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.0514 - 1.07i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-7.06 + 15.4i)T + (-63.5 - 73.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.09967959277972569628580057874, −10.62144804932149201832006657382, −9.881756001214476576167467582247, −9.157211259268174154410349627794, −8.333257263188318526422486588345, −6.63945766368292193641213993728, −5.01317982192455029966815748555, −4.62264322926476517813623286995, −3.23008375682808407415396694444, −2.02144027769149298933766078621, 1.88841943429580962548641056322, 2.70537402116415291192064019723, 4.86967066611701757460091445033, 6.13254400474353271120562747483, 6.57035222296626309805407331773, 7.82649071833721732396576480433, 8.608764010200608787706113028728, 9.262421816844550127043643089337, 10.70516944476350232212647903326, 11.99181281610956392581727461326

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