Properties

Label 2-322-23.3-c1-0-4
Degree $2$
Conductor $322$
Sign $-0.471 - 0.881i$
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  + (0.142 + 0.989i)2-s + (0.527 − 1.15i)3-s + (−0.959 + 0.281i)4-s + (−2.43 + 2.81i)5-s + (1.21 + 0.357i)6-s + (0.841 − 0.540i)7-s + (−0.415 − 0.909i)8-s + (0.909 + 1.05i)9-s + (−3.12 − 2.01i)10-s + (−0.680 + 4.73i)11-s + (−0.180 + 1.25i)12-s + (0.0700 + 0.0450i)13-s + (0.654 + 0.755i)14-s + (1.96 + 4.29i)15-s + (0.841 − 0.540i)16-s + (−0.508 − 0.149i)17-s + ⋯
L(s)  = 1  + (0.100 + 0.699i)2-s + (0.304 − 0.666i)3-s + (−0.479 + 0.140i)4-s + (−1.08 + 1.25i)5-s + (0.497 + 0.145i)6-s + (0.317 − 0.204i)7-s + (−0.146 − 0.321i)8-s + (0.303 + 0.350i)9-s + (−0.989 − 0.635i)10-s + (−0.205 + 1.42i)11-s + (−0.0521 + 0.362i)12-s + (0.0194 + 0.0124i)13-s + (0.175 + 0.201i)14-s + (0.506 + 1.10i)15-s + (0.210 − 0.135i)16-s + (−0.123 − 0.0362i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.471 - 0.881i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.471 - 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.565138 + 0.943218i\)
\(L(\frac12)\) \(\approx\) \(0.565138 + 0.943218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 - 0.989i)T \)
7 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (-3.48 - 3.29i)T \)
good3 \( 1 + (-0.527 + 1.15i)T + (-1.96 - 2.26i)T^{2} \)
5 \( 1 + (2.43 - 2.81i)T + (-0.711 - 4.94i)T^{2} \)
11 \( 1 + (0.680 - 4.73i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (-0.0700 - 0.0450i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (0.508 + 0.149i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (7.88 - 2.31i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-4.70 - 1.38i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-3.56 - 7.81i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (5.30 + 6.12i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (-4.87 + 5.62i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (-3.85 + 8.43i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 - 2.50T + 47T^{2} \)
53 \( 1 + (-3.86 + 2.48i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (12.0 + 7.71i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (2.87 + 6.28i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-0.996 - 6.93i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (-1.58 - 11.0i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-12.0 + 3.53i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (-6.59 - 4.23i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (4.70 + 5.42i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (0.539 - 1.18i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (3.94 - 4.55i)T + (-13.8 - 96.0i)T^{2} \)
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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

−12.25162245218573231536644002580, −10.81596403337500966867764806976, −10.32343408059848574415205184632, −8.698754151567537186482760283614, −7.77181461988788694386406576874, −7.15345779507326582877403279448, −6.63809751841817364673971165246, −4.83507058882153913798912346186, −3.81744120087739813238670477678, −2.24394862192726321495300398899, 0.77071461014865079698051701731, 2.97875830720643996068797338945, 4.28733929712051931114911765946, 4.65110359009997262912858248237, 6.21244524510351587748025149389, 8.077317615309091071842091775873, 8.626056386861629766992558423985, 9.277868729645173506587319108529, 10.60740279710607252765238109034, 11.28099074409281570528078511334

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