Properties

Label 2-320-320.187-c1-0-31
Degree $2$
Conductor $320$
Sign $-0.721 + 0.692i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
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  + (−1.32 − 0.506i)2-s + (0.625 + 3.14i)3-s + (1.48 + 1.33i)4-s + (−2.02 − 0.949i)5-s + (0.766 − 4.47i)6-s + (−1.61 − 3.90i)7-s + (−1.28 − 2.51i)8-s + (−6.73 + 2.79i)9-s + (2.19 + 2.27i)10-s + (−1.28 + 0.858i)11-s + (−3.27 + 5.51i)12-s + (−0.465 + 0.696i)13-s + (0.159 + 5.96i)14-s + (1.72 − 6.96i)15-s + (0.425 + 3.97i)16-s − 3.39·17-s + ⋯
L(s)  = 1  + (−0.933 − 0.357i)2-s + (0.361 + 1.81i)3-s + (0.743 + 0.668i)4-s + (−0.905 − 0.424i)5-s + (0.312 − 1.82i)6-s + (−0.610 − 1.47i)7-s + (−0.455 − 0.890i)8-s + (−2.24 + 0.930i)9-s + (0.693 + 0.720i)10-s + (−0.387 + 0.258i)11-s + (−0.945 + 1.59i)12-s + (−0.129 + 0.193i)13-s + (0.0425 + 1.59i)14-s + (0.444 − 1.79i)15-s + (0.106 + 0.994i)16-s − 0.823·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.721 + 0.692i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.721 + 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00667753 - 0.0165847i\)
\(L(\frac12)\) \(\approx\) \(0.00667753 - 0.0165847i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 0.506i)T \)
5 \( 1 + (2.02 + 0.949i)T \)
good3 \( 1 + (-0.625 - 3.14i)T + (-2.77 + 1.14i)T^{2} \)
7 \( 1 + (1.61 + 3.90i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.28 - 0.858i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + (0.465 - 0.696i)T + (-4.97 - 12.0i)T^{2} \)
17 \( 1 + 3.39T + 17T^{2} \)
19 \( 1 + (0.673 + 3.38i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (6.81 + 2.82i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-5.12 - 3.42i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 - 1.40T + 31T^{2} \)
37 \( 1 + (0.828 - 0.553i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (2.83 - 1.17i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (0.308 + 0.0614i)T + (39.7 + 16.4i)T^{2} \)
47 \( 1 - 5.50iT - 47T^{2} \)
53 \( 1 + (9.36 + 1.86i)T + (48.9 + 20.2i)T^{2} \)
59 \( 1 + (-1.53 - 0.305i)T + (54.5 + 22.5i)T^{2} \)
61 \( 1 + (-4.84 - 3.23i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (-3.91 + 0.778i)T + (61.8 - 25.6i)T^{2} \)
71 \( 1 + (14.8 + 6.14i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (0.991 + 0.410i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (3.04 + 3.04i)T + 79iT^{2} \)
83 \( 1 + (2.86 - 4.28i)T + (-31.7 - 76.6i)T^{2} \)
89 \( 1 + (2.13 - 5.15i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 + (6.91 - 6.91i)T - 97iT^{2} \)
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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

−10.82400920453435197633389565255, −10.34374657608840424699806065371, −9.562913030325887645193382608449, −8.680200954778334155506558018900, −7.87013044728269470680570745566, −6.69276230245301501298751802799, −4.61526658685145194168351747900, −4.02234397591838909649326379295, −2.97351326172393409942665791739, −0.01521264560537566024821270862, 2.07181863930818142567644156118, 2.98340148623043855059832439163, 5.79328680178115308586853334042, 6.41323500252322528424614773654, 7.34346570584916700926810920264, 8.292180011657516799888937477345, 8.563732717232049743789821559349, 9.923327167477166466381707773486, 11.35974360398455189734230458100, 11.99062985578187794606184763762

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