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Properties

Label 2-322-161.10-c1-0-7
Degree $2$
Conductor $322$
Sign $0.776 - 0.629i$
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 + (0.451 + 0.876i)3-s + (−0.786 − 0.618i)4-s + (2.24 − 0.214i)5-s + (−0.976 + 0.140i)6-s + (0.164 − 2.64i)7-s + (0.841 − 0.540i)8-s + (1.17 − 1.65i)9-s + (−0.530 + 2.18i)10-s + (3.58 − 1.24i)11-s + (0.186 − 0.968i)12-s + (0.694 + 2.36i)13-s + (2.44 + 1.01i)14-s + (1.20 + 1.86i)15-s + (0.235 + 0.971i)16-s + (−1.51 − 0.605i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.668i)2-s + (0.260 + 0.506i)3-s + (−0.393 − 0.309i)4-s + (1.00 − 0.0957i)5-s + (−0.398 + 0.0573i)6-s + (0.0621 − 0.998i)7-s + (0.297 − 0.191i)8-s + (0.391 − 0.550i)9-s + (−0.167 + 0.691i)10-s + (1.08 − 0.374i)11-s + (0.0538 − 0.279i)12-s + (0.192 + 0.655i)13-s + (0.652 + 0.272i)14-s + (0.309 + 0.482i)15-s + (0.0589 + 0.242i)16-s + (−0.366 − 0.146i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 - 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.776 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.776 - 0.629i$
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: \((2,\ 322,\ (\ :1/2),\ 0.776 - 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41767 + 0.502450i\)
\(L(\frac12)\) \(\approx\) \(1.41767 + 0.502450i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.164 + 2.64i)T \)
23 \( 1 + (4.41 - 1.87i)T \)
good3 \( 1 + (-0.451 - 0.876i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (-2.24 + 0.214i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-3.58 + 1.24i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.694 - 2.36i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (1.51 + 0.605i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (3.39 - 1.35i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.134 - 0.937i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (2.63 - 0.125i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-7.04 - 5.01i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-3.54 - 1.61i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (2.10 - 3.26i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (3.67 - 2.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.11 + 3.26i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (10.5 + 2.54i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-12.2 - 6.32i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (2.01 + 10.4i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (5.11 + 5.89i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (1.34 - 1.71i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-6.97 + 7.31i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-1.07 - 2.36i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.0713 - 1.49i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (5.95 - 13.0i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.56749246672231851605738444679, −10.46972907402515986085795364777, −9.597097132263212392221596287402, −9.177088226892134578972283365451, −7.971485427690386167944450654791, −6.64295746814281002920549064134, −6.18112223905620805291907888624, −4.57775699406953769762874602885, −3.75035157449713582193978825354, −1.49148504557172277071249226470, 1.76372411413059945881046810314, 2.50062837663319769645085871573, 4.25518947946657895771330938305, 5.65594024836382875227135736785, 6.61750864816316541930325645473, 7.953218513979853887754385593741, 8.863114259003346973875475435128, 9.657687088996873793139830074183, 10.53021721745362351403609964335, 11.55066131249213397910032915282

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