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Properties

Label 2-322-7.2-c1-0-4
Degree $2$
Conductor $322$
Sign $0.452 - 0.891i$
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  + (0.5 − 0.866i)2-s + (1.43 + 2.47i)3-s + (−0.499 − 0.866i)4-s + (−0.686 + 1.18i)5-s + 2.86·6-s + (−2.30 + 1.30i)7-s − 0.999·8-s + (−2.59 + 4.49i)9-s + (0.686 + 1.18i)10-s + (1.21 + 2.09i)11-s + (1.43 − 2.47i)12-s + 5.67·13-s + (−0.0242 + 2.64i)14-s − 3.92·15-s + (−0.5 + 0.866i)16-s + (−2.18 − 3.78i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.825 + 1.43i)3-s + (−0.249 − 0.433i)4-s + (−0.306 + 0.531i)5-s + 1.16·6-s + (−0.870 + 0.492i)7-s − 0.353·8-s + (−0.864 + 1.49i)9-s + (0.217 + 0.375i)10-s + (0.365 + 0.632i)11-s + (0.412 − 0.715i)12-s + 1.57·13-s + (−0.00649 + 0.707i)14-s − 1.01·15-s + (−0.125 + 0.216i)16-s + (−0.530 − 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.452 - 0.891i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.452 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48804 + 0.913748i\)
\(L(\frac12)\) \(\approx\) \(1.48804 + 0.913748i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.30 - 1.30i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-1.43 - 2.47i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.686 - 1.18i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.21 - 2.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.67T + 13T^{2} \)
17 \( 1 + (2.18 + 3.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.735 + 1.27i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 3.06T + 29T^{2} \)
31 \( 1 + (3.65 + 6.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.13 + 3.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 + 2.39T + 43T^{2} \)
47 \( 1 + (-0.915 + 1.58i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.12 + 3.68i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.416 + 0.721i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.67 + 13.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.79 - 8.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + (-2.22 - 3.86i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.50 - 12.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.16T + 83T^{2} \)
89 \( 1 + (8.98 - 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.26T + 97T^{2} \)
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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

−11.40531366331629049481861976806, −10.93859529364398490918554282903, −9.763327826998129573195446055108, −9.351943274404393252191847751118, −8.459470299403205130240438978395, −6.87692511449351201127971062642, −5.57155911499524972515155184577, −4.24249903856706000876070794589, −3.51459032865234843216523432358, −2.58045569337166277537975048403, 1.16188430223833592435471202130, 3.11638550737014485255222719456, 4.08349466336572771394109098486, 6.08188936638721235371383349095, 6.53765972596996759684635181027, 7.61915582319037313827355980119, 8.559410879039267353991718167236, 8.899439248671651676093187966556, 10.60138310484529215211584272962, 11.90805993895177030583030077181

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