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Properties

Label 2-322-161.10-c1-0-2
Degree $2$
Conductor $322$
Sign $-0.135 - 0.990i$
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 + (1.37 + 2.66i)3-s + (−0.786 − 0.618i)4-s + (−4.34 + 0.415i)5-s + (2.97 − 0.427i)6-s + (2.28 + 1.33i)7-s + (−0.841 + 0.540i)8-s + (−3.48 + 4.89i)9-s + (−1.02 + 4.24i)10-s + (−3.38 + 1.17i)11-s + (0.567 − 2.94i)12-s + (0.516 + 1.75i)13-s + (2.00 − 1.72i)14-s + (−7.08 − 11.0i)15-s + (0.235 + 0.971i)16-s + (3.05 + 1.22i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (0.793 + 1.53i)3-s + (−0.393 − 0.309i)4-s + (−1.94 + 0.185i)5-s + (1.21 − 0.174i)6-s + (0.864 + 0.502i)7-s + (−0.297 + 0.191i)8-s + (−1.16 + 1.63i)9-s + (−0.325 + 1.34i)10-s + (−1.02 + 0.353i)11-s + (0.163 − 0.850i)12-s + (0.143 + 0.487i)13-s + (0.535 − 0.461i)14-s + (−1.82 − 2.84i)15-s + (0.0589 + 0.242i)16-s + (0.740 + 0.296i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.135 - 0.990i$
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: \((2,\ 322,\ (\ :1/2),\ -0.135 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.791304 + 0.906577i\)
\(L(\frac12)\) \(\approx\) \(0.791304 + 0.906577i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.28 - 1.33i)T \)
23 \( 1 + (4.55 - 1.50i)T \)
good3 \( 1 + (-1.37 - 2.66i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (4.34 - 0.415i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (3.38 - 1.17i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.516 - 1.75i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-3.05 - 1.22i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-2.87 + 1.15i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.135 + 0.944i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-0.781 + 0.0372i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (1.24 + 0.883i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-8.45 - 3.85i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (2.61 - 4.06i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-10.0 + 5.79i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.53 + 2.65i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-4.72 - 1.14i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-0.0381 - 0.0196i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (2.13 + 11.0i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-0.761 - 0.879i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (2.81 - 3.58i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-2.46 + 2.58i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-2.50 - 5.48i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.283 - 5.95i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-3.30 + 7.23i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.62551088652303011605558784170, −10.99431181487431403214508849151, −10.20100316506017234912316294135, −9.127222971715381950736046642470, −8.175642074192235259781587682759, −7.69741613585664230559238536678, −5.27532010169576290472264135318, −4.43466849474990109702114737307, −3.71957960195280424522228374611, −2.68490596533345192709199965969, 0.76595648445171456264521668789, 2.97879478154624692630821219335, 4.08527786395090486078760119007, 5.50021445031345257566247082809, 7.07495013720239939469453830332, 7.82889985286970996551796226927, 7.88072406369076184992506057597, 8.755290883892642092146274267266, 10.69631405026698231825343800061, 11.83858262894312007130937071593

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