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Properties

Label 2-322-161.10-c1-0-1
Degree $2$
Conductor $322$
Sign $-0.718 + 0.695i$
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 + (0.958 + 1.85i)3-s + (−0.786 − 0.618i)4-s + (−3.50 + 0.334i)5-s + (−2.07 + 0.297i)6-s + (−2.57 + 0.620i)7-s + (0.841 − 0.540i)8-s + (−0.799 + 1.12i)9-s + (0.829 − 3.42i)10-s + (0.598 − 0.207i)11-s + (0.396 − 2.05i)12-s + (−0.825 − 2.81i)13-s + (0.254 − 2.63i)14-s + (−3.98 − 6.19i)15-s + (0.235 + 0.971i)16-s + (0.623 + 0.249i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.668i)2-s + (0.553 + 1.07i)3-s + (−0.393 − 0.309i)4-s + (−1.56 + 0.149i)5-s + (−0.845 + 0.121i)6-s + (−0.972 + 0.234i)7-s + (0.297 − 0.191i)8-s + (−0.266 + 0.374i)9-s + (0.262 − 1.08i)10-s + (0.180 − 0.0624i)11-s + (0.114 − 0.593i)12-s + (−0.229 − 0.780i)13-s + (0.0679 − 0.703i)14-s + (−1.02 − 1.59i)15-s + (0.0589 + 0.242i)16-s + (0.151 + 0.0605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.718 + 0.695i$
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: \((2,\ 322,\ (\ :1/2),\ -0.718 + 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153402 - 0.379347i\)
\(L(\frac12)\) \(\approx\) \(0.153402 - 0.379347i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.57 - 0.620i)T \)
23 \( 1 + (-0.903 - 4.70i)T \)
good3 \( 1 + (-0.958 - 1.85i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (3.50 - 0.334i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-0.598 + 0.207i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (0.825 + 2.81i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.623 - 0.249i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (6.38 - 2.55i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.855 - 5.94i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (9.36 - 0.446i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (1.07 + 0.765i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (4.06 + 1.85i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (3.78 - 5.88i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (5.47 - 3.16i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.07 - 2.18i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-11.7 - 2.84i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (6.79 + 3.50i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (0.761 + 3.94i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-1.93 - 2.23i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-7.19 + 9.15i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (1.36 - 1.43i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-4.80 - 10.5i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.157 + 3.31i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (3.26 - 7.15i)T + (-63.5 - 73.3i)T^{2} \)
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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

−12.28825848645613502206223120825, −10.98091909501949117914443020295, −10.21773997964230480402811099198, −9.231865217855281709971411542785, −8.496788169558390912338390945763, −7.58656599495549594968497033992, −6.56043921774777099870731849073, −5.13290184700193319255207462220, −3.86190431027677858367777179664, −3.34285789926770301636681489082, 0.28745476592529387600869729902, 2.22187580600541991898588639074, 3.54975711620064066880981772465, 4.47501057587741090008517453522, 6.67268205285893205004677935095, 7.25282032289361610566372214947, 8.306139760219416844541332281235, 8.898319239227701785794185820262, 10.19810257782412657986331468356, 11.27562590315583251156989848462

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