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Properties

Label 2-322-161.103-c1-0-10
Degree $2$
Conductor $322$
Sign $0.992 - 0.120i$
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  + (0.0475 + 0.998i)2-s + (−0.0852 − 0.212i)3-s + (−0.995 + 0.0950i)4-s + (0.0734 − 0.302i)5-s + (0.208 − 0.0952i)6-s + (1.11 − 2.39i)7-s + (−0.142 − 0.989i)8-s + (2.13 − 2.03i)9-s + (0.305 + 0.0589i)10-s + (1.99 + 0.0950i)11-s + (0.105 + 0.203i)12-s + (−0.854 − 0.740i)13-s + (2.44 + 1.00i)14-s + (−0.0706 + 0.0101i)15-s + (0.981 − 0.189i)16-s + (−0.160 + 0.225i)17-s + ⋯
L(s)  = 1  + (0.0336 + 0.706i)2-s + (−0.0492 − 0.122i)3-s + (−0.497 + 0.0475i)4-s + (0.0328 − 0.135i)5-s + (0.0851 − 0.0388i)6-s + (0.422 − 0.906i)7-s + (−0.0503 − 0.349i)8-s + (0.711 − 0.677i)9-s + (0.0967 + 0.0186i)10-s + (0.601 + 0.0286i)11-s + (0.0303 + 0.0588i)12-s + (−0.237 − 0.205i)13-s + (0.654 + 0.268i)14-s + (−0.0182 + 0.00262i)15-s + (0.245 − 0.0473i)16-s + (−0.0389 + 0.0546i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.992 - 0.120i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.992 - 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37746 + 0.0829767i\)
\(L(\frac12)\) \(\approx\) \(1.37746 + 0.0829767i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0475 - 0.998i)T \)
7 \( 1 + (-1.11 + 2.39i)T \)
23 \( 1 + (2.39 + 4.15i)T \)
good3 \( 1 + (0.0852 + 0.212i)T + (-2.17 + 2.07i)T^{2} \)
5 \( 1 + (-0.0734 + 0.302i)T + (-4.44 - 2.29i)T^{2} \)
11 \( 1 + (-1.99 - 0.0950i)T + (10.9 + 1.04i)T^{2} \)
13 \( 1 + (0.854 + 0.740i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (0.160 - 0.225i)T + (-5.56 - 16.0i)T^{2} \)
19 \( 1 + (-4.29 - 6.02i)T + (-6.21 + 17.9i)T^{2} \)
29 \( 1 + (1.98 + 4.34i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-2.90 - 3.69i)T + (-7.30 + 30.1i)T^{2} \)
37 \( 1 + (3.03 + 3.18i)T + (-1.76 + 36.9i)T^{2} \)
41 \( 1 + (0.623 + 2.12i)T + (-34.4 + 22.1i)T^{2} \)
43 \( 1 + (3.16 + 0.455i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (-1.92 - 1.11i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.698 - 0.241i)T + (41.6 + 32.7i)T^{2} \)
59 \( 1 + (2.78 - 14.4i)T + (-54.7 - 21.9i)T^{2} \)
61 \( 1 + (5.92 + 2.37i)T + (44.1 + 42.0i)T^{2} \)
67 \( 1 + (0.603 - 1.16i)T + (-38.8 - 54.5i)T^{2} \)
71 \( 1 + (7.54 - 4.84i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-0.748 - 7.83i)T + (-71.6 + 13.8i)T^{2} \)
79 \( 1 + (-8.22 + 2.84i)T + (62.0 - 48.8i)T^{2} \)
83 \( 1 + (8.30 + 2.43i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (4.87 + 3.83i)T + (20.9 + 86.4i)T^{2} \)
97 \( 1 + (-14.0 + 4.13i)T + (81.6 - 52.4i)T^{2} \)
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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

−11.86546886712289633534589552835, −10.47897097829741911463481183922, −9.783041762334475761358595691412, −8.687428938349578151466287169168, −7.62038260717928043217441773370, −6.93103454067749029104183685165, −5.86706949949003582874707790102, −4.56107033339126366626783088454, −3.65191921427009573988484758781, −1.22016039384723690590257131106, 1.70267510786328894192786583631, 3.02080265207529289808946301420, 4.54147518164221365794287321958, 5.32825147183451886660396891450, 6.78907203344518482057722283761, 7.944633272151671540985228987158, 9.073908376438940859830790777159, 9.722028111389229594827484249002, 10.84930243430034275999620552508, 11.57834632972212153168218461633

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