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Properties

Label 2-322-161.18-c1-0-1
Degree $2$
Conductor $322$
Sign $0.777 - 0.628i$
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  + (−0.235 − 0.971i)2-s + (−0.457 − 1.32i)3-s + (−0.888 + 0.458i)4-s + (−4.01 + 1.60i)5-s + (−1.17 + 0.755i)6-s + (1.65 + 2.06i)7-s + (0.654 + 0.755i)8-s + (0.822 − 0.646i)9-s + (2.50 + 3.51i)10-s + (−0.428 + 1.76i)11-s + (1.01 + 0.964i)12-s + (0.729 − 1.59i)13-s + (1.61 − 2.09i)14-s + (3.95 + 4.56i)15-s + (0.580 − 0.814i)16-s + (0.363 + 7.62i)17-s + ⋯
L(s)  = 1  + (−0.166 − 0.687i)2-s + (−0.263 − 0.762i)3-s + (−0.444 + 0.229i)4-s + (−1.79 + 0.718i)5-s + (−0.480 + 0.308i)6-s + (0.624 + 0.781i)7-s + (0.231 + 0.267i)8-s + (0.274 − 0.215i)9-s + (0.792 + 1.11i)10-s + (−0.129 + 0.532i)11-s + (0.292 + 0.278i)12-s + (0.202 − 0.442i)13-s + (0.432 − 0.559i)14-s + (1.02 + 1.17i)15-s + (0.145 − 0.203i)16-s + (0.0880 + 1.84i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.777 - 0.628i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.777 - 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.579218 + 0.204866i\)
\(L(\frac12)\) \(\approx\) \(0.579218 + 0.204866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.235 + 0.971i)T \)
7 \( 1 + (-1.65 - 2.06i)T \)
23 \( 1 + (0.236 - 4.79i)T \)
good3 \( 1 + (0.457 + 1.32i)T + (-2.35 + 1.85i)T^{2} \)
5 \( 1 + (4.01 - 1.60i)T + (3.61 - 3.45i)T^{2} \)
11 \( 1 + (0.428 - 1.76i)T + (-9.77 - 5.04i)T^{2} \)
13 \( 1 + (-0.729 + 1.59i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-0.363 - 7.62i)T + (-16.9 + 1.61i)T^{2} \)
19 \( 1 + (0.124 - 2.62i)T + (-18.9 - 1.80i)T^{2} \)
29 \( 1 + (1.63 - 1.05i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (4.47 - 0.863i)T + (28.7 - 11.5i)T^{2} \)
37 \( 1 + (2.79 - 2.19i)T + (8.72 - 35.9i)T^{2} \)
41 \( 1 + (0.955 - 6.64i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-3.75 + 4.32i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (-0.638 - 1.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.391 - 0.0373i)T + (52.0 + 10.0i)T^{2} \)
59 \( 1 + (0.825 + 1.15i)T + (-19.2 + 55.7i)T^{2} \)
61 \( 1 + (2.56 - 7.40i)T + (-47.9 - 37.7i)T^{2} \)
67 \( 1 + (7.04 - 6.71i)T + (3.18 - 66.9i)T^{2} \)
71 \( 1 + (-5.41 - 1.59i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-3.90 + 2.01i)T + (42.3 - 59.4i)T^{2} \)
79 \( 1 + (4.09 - 0.391i)T + (77.5 - 14.9i)T^{2} \)
83 \( 1 + (1.82 + 12.7i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (9.26 + 1.78i)T + (82.6 + 33.0i)T^{2} \)
97 \( 1 + (-2.33 + 16.2i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.78110714651252147822420816788, −11.05744007370358052379370651843, −10.20120670901227616205903726039, −8.653533097745479696791138120602, −7.891237278775266802176923307459, −7.24772838069668093040254533894, −5.89607828581543623176594800377, −4.27801513149997913099456682184, −3.35790221415885388810956738118, −1.66374482764527338516650235055, 0.51362541413016146905356883967, 3.73244569422239589883366189421, 4.57082254616477742495136924587, 5.13116783176445525966811413451, 7.08924091146442606298547952212, 7.62161706057158792305733047430, 8.590892009119400562840642569733, 9.437143296807011461064927993676, 10.86872548233950124566081910800, 11.23681869243739543350035470986

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