Properties

Label 2-322-161.97-c1-0-8
Degree $2$
Conductor $322$
Sign $-0.666 + 0.745i$
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  + (−0.959 − 0.281i)2-s + (−0.749 − 0.649i)3-s + (0.841 + 0.540i)4-s + (−0.540 + 3.76i)5-s + (0.536 + 0.834i)6-s + (−2.61 + 0.378i)7-s + (−0.654 − 0.755i)8-s + (−0.286 − 1.99i)9-s + (1.57 − 3.45i)10-s + (−1.64 − 5.59i)11-s + (−0.279 − 0.951i)12-s + (0.519 + 0.237i)13-s + (2.61 + 0.374i)14-s + (2.84 − 2.46i)15-s + (0.415 + 0.909i)16-s + (5.39 − 3.46i)17-s + ⋯
L(s)  = 1  + (−0.678 − 0.199i)2-s + (−0.432 − 0.375i)3-s + (0.420 + 0.270i)4-s + (−0.241 + 1.68i)5-s + (0.218 + 0.340i)6-s + (−0.989 + 0.143i)7-s + (−0.231 − 0.267i)8-s + (−0.0956 − 0.665i)9-s + (0.499 − 1.09i)10-s + (−0.495 − 1.68i)11-s + (−0.0806 − 0.274i)12-s + (0.144 + 0.0658i)13-s + (0.699 + 0.100i)14-s + (0.735 − 0.637i)15-s + (0.103 + 0.227i)16-s + (1.30 − 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.666 + 0.745i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.666 + 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128381 - 0.287061i\)
\(L(\frac12)\) \(\approx\) \(0.128381 - 0.287061i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (2.61 - 0.378i)T \)
23 \( 1 + (-2.13 + 4.29i)T \)
good3 \( 1 + (0.749 + 0.649i)T + (0.426 + 2.96i)T^{2} \)
5 \( 1 + (0.540 - 3.76i)T + (-4.79 - 1.40i)T^{2} \)
11 \( 1 + (1.64 + 5.59i)T + (-9.25 + 5.94i)T^{2} \)
13 \( 1 + (-0.519 - 0.237i)T + (8.51 + 9.82i)T^{2} \)
17 \( 1 + (-5.39 + 3.46i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (5.54 + 3.56i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-0.651 + 0.418i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (5.47 - 4.74i)T + (4.41 - 30.6i)T^{2} \)
37 \( 1 + (0.936 - 0.134i)T + (35.5 - 10.4i)T^{2} \)
41 \( 1 + (2.88 + 0.415i)T + (39.3 + 11.5i)T^{2} \)
43 \( 1 + (-1.26 - 1.10i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 - 1.08iT - 47T^{2} \)
53 \( 1 + (3.78 - 1.72i)T + (34.7 - 40.0i)T^{2} \)
59 \( 1 + (8.99 + 4.10i)T + (38.6 + 44.5i)T^{2} \)
61 \( 1 + (4.78 + 5.52i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (1.75 - 5.98i)T + (-56.3 - 36.2i)T^{2} \)
71 \( 1 + (-9.07 - 2.66i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (2.13 - 3.31i)T + (-30.3 - 66.4i)T^{2} \)
79 \( 1 + (3.57 + 1.63i)T + (51.7 + 59.7i)T^{2} \)
83 \( 1 + (1.38 + 9.63i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-5.73 + 6.62i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (0.176 - 1.22i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.98972604279547195175122805227, −10.66419367625383731649872367203, −9.515910846537483649425181516139, −8.491078873309689246371521149851, −7.22723652171525992473245904865, −6.55944003060731314782070269702, −5.86487141501043878061786449261, −3.35640883046443745990323729554, −2.90199930654795190143217664036, −0.28493394778369751797263518081, 1.74438979389488063674637427696, 4.00671596997136985062404372915, 5.09395230606645246415857302701, 5.94682832102894729689673801518, 7.49085026046423784771160949543, 8.167815953885235318445418824886, 9.315515311152786121530501230073, 9.960430900961108364159972678721, 10.72405488074438484112603995384, 12.19874477888736549873155994666

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