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Properties

Label 2-322-161.100-c1-0-10
Degree $2$
Conductor $322$
Sign $0.725 - 0.687i$
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  + (0.786 + 0.618i)2-s + (−0.191 + 0.269i)3-s + (0.235 + 0.971i)4-s + (1.44 − 0.279i)5-s + (−0.317 + 0.0930i)6-s + (1.35 − 2.27i)7-s + (−0.415 + 0.909i)8-s + (0.945 + 2.73i)9-s + (1.31 + 0.676i)10-s + (1.04 − 0.825i)11-s + (−0.306 − 0.122i)12-s + (−1.50 + 0.965i)13-s + (2.47 − 0.946i)14-s + (−0.202 + 0.443i)15-s + (−0.888 + 0.458i)16-s + (−0.110 − 0.105i)17-s + ⋯
L(s)  = 1  + (0.555 + 0.437i)2-s + (−0.110 + 0.155i)3-s + (0.117 + 0.485i)4-s + (0.648 − 0.124i)5-s + (−0.129 + 0.0380i)6-s + (0.512 − 0.858i)7-s + (−0.146 + 0.321i)8-s + (0.315 + 0.910i)9-s + (0.415 + 0.213i)10-s + (0.316 − 0.248i)11-s + (−0.0885 − 0.0354i)12-s + (−0.416 + 0.267i)13-s + (0.660 − 0.252i)14-s + (−0.0523 + 0.114i)15-s + (−0.222 + 0.114i)16-s + (−0.0269 − 0.0256i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.725 - 0.687i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.725 - 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81375 + 0.723009i\)
\(L(\frac12)\) \(\approx\) \(1.81375 + 0.723009i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.786 - 0.618i)T \)
7 \( 1 + (-1.35 + 2.27i)T \)
23 \( 1 + (-3.64 - 3.11i)T \)
good3 \( 1 + (0.191 - 0.269i)T + (-0.981 - 2.83i)T^{2} \)
5 \( 1 + (-1.44 + 0.279i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (-1.04 + 0.825i)T + (2.59 - 10.6i)T^{2} \)
13 \( 1 + (1.50 - 0.965i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (0.110 + 0.105i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (-1.80 + 1.72i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (5.22 - 1.53i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (1.81 - 0.173i)T + (30.4 - 5.86i)T^{2} \)
37 \( 1 + (0.724 + 2.09i)T + (-29.0 + 22.8i)T^{2} \)
41 \( 1 + (4.77 + 5.50i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-0.773 - 1.69i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 + (-5.52 + 9.56i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.244 + 5.12i)T + (-52.7 - 5.03i)T^{2} \)
59 \( 1 + (-0.0545 - 0.0281i)T + (34.2 + 48.0i)T^{2} \)
61 \( 1 + (7.74 + 10.8i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (9.32 - 3.73i)T + (48.4 - 46.2i)T^{2} \)
71 \( 1 + (0.819 - 5.69i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-1.43 - 5.90i)T + (-64.8 + 33.4i)T^{2} \)
79 \( 1 + (0.0127 + 0.268i)T + (-78.6 + 7.50i)T^{2} \)
83 \( 1 + (-3.55 + 4.10i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (8.54 + 0.815i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (-0.616 - 0.711i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.65929460504856213794996134840, −10.91613974296536669020291384830, −9.936166727007849287908718945712, −8.892701831047585203109995848116, −7.60624891183493909749476961071, −7.01724575290765740210550494177, −5.58029286588814874038365490856, −4.87013564922540175793066375810, −3.69235163281320204029631205616, −1.88266232927323043066715739930, 1.62551149163675024272692310915, 2.94314711312247235228971221015, 4.40816837517062869964272093136, 5.59276685609629263635718380495, 6.32991587105950553638565547995, 7.55768640210360471611510446081, 9.015739520084408752667809438302, 9.658717085824558473801688617005, 10.69470701929418393614516620481, 11.78778962278922830172042373869

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