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Properties

Label 2-320-80.13-c2-0-19
Degree $2$
Conductor $320$
Sign $-0.978 + 0.206i$
Analytic cond. $8.71936$
Root an. cond. $2.95285$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.73i·3-s + (4.99 + 0.266i)5-s + (−3.79 − 3.79i)7-s − 23.8·9-s + (2.45 − 2.45i)11-s − 11.2i·13-s + (1.52 − 28.6i)15-s + (−1.15 + 1.15i)17-s + (−10.7 + 10.7i)19-s + (−21.7 + 21.7i)21-s + (16.0 − 16.0i)23-s + (24.8 + 2.65i)25-s + 85.2i·27-s + (10.2 − 10.2i)29-s − 30.2·31-s + ⋯
L(s)  = 1  − 1.91i·3-s + (0.998 + 0.0532i)5-s + (−0.542 − 0.542i)7-s − 2.65·9-s + (0.223 − 0.223i)11-s − 0.868i·13-s + (0.101 − 1.90i)15-s + (−0.0678 + 0.0678i)17-s + (−0.564 + 0.564i)19-s + (−1.03 + 1.03i)21-s + (0.697 − 0.697i)23-s + (0.994 + 0.106i)25-s + 3.15i·27-s + (0.352 − 0.352i)29-s − 0.976·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.978 + 0.206i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -0.978 + 0.206i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.154052 - 1.47569i\)
\(L(\frac12)\) \(\approx\) \(0.154052 - 1.47569i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-4.99 - 0.266i)T \)
good3 \( 1 + 5.73iT - 9T^{2} \)
7 \( 1 + (3.79 + 3.79i)T + 49iT^{2} \)
11 \( 1 + (-2.45 + 2.45i)T - 121iT^{2} \)
13 \( 1 + 11.2iT - 169T^{2} \)
17 \( 1 + (1.15 - 1.15i)T - 289iT^{2} \)
19 \( 1 + (10.7 - 10.7i)T - 361iT^{2} \)
23 \( 1 + (-16.0 + 16.0i)T - 529iT^{2} \)
29 \( 1 + (-10.2 + 10.2i)T - 841iT^{2} \)
31 \( 1 + 30.2T + 961T^{2} \)
37 \( 1 - 18.0iT - 1.36e3T^{2} \)
41 \( 1 - 18.0iT - 1.68e3T^{2} \)
43 \( 1 + 45.3T + 1.84e3T^{2} \)
47 \( 1 + (-14.1 + 14.1i)T - 2.20e3iT^{2} \)
53 \( 1 - 46.7T + 2.80e3T^{2} \)
59 \( 1 + (-6.15 - 6.15i)T + 3.48e3iT^{2} \)
61 \( 1 + (20.5 + 20.5i)T + 3.72e3iT^{2} \)
67 \( 1 - 108.T + 4.48e3T^{2} \)
71 \( 1 + 7.36iT - 5.04e3T^{2} \)
73 \( 1 + (-84.7 + 84.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 42.0iT - 6.24e3T^{2} \)
83 \( 1 - 10.5iT - 6.88e3T^{2} \)
89 \( 1 + 14.0T + 7.92e3T^{2} \)
97 \( 1 + (69.9 - 69.9i)T - 9.40e3iT^{2} \)
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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

−11.06934307008794080933168211918, −10.13538280316663620503876444400, −8.849286389561443726918140216769, −8.000390412620031533077395517273, −6.88665302734356597576550784530, −6.35671084678792588039574214221, −5.40491751950040051895792972550, −3.16904264829344078629038993878, −1.98067729657109191025688282412, −0.68737666330785780691148953553, 2.45652577182369444316822179025, 3.69934225925180168001401807006, 4.86660723192502108839548867038, 5.64670534075935149705804880517, 6.72061042499301302377786478228, 8.753472951588590010896141310889, 9.216540373391515854467762017806, 9.831833736308098790064262296463, 10.73573191701864052602421052547, 11.50452864080070496197477725512

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