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Properties

Label 2-320-16.3-c2-0-12
Degree $2$
Conductor $320$
Sign $-0.612 + 0.790i$
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  + (−1.92 + 1.92i)3-s + (1.58 − 1.58i)5-s − 4.10·7-s + 1.58i·9-s + (0.338 + 0.338i)11-s + (−11.8 − 11.8i)13-s + 6.09i·15-s − 9.17·17-s + (5.20 − 5.20i)19-s + (7.91 − 7.91i)21-s − 5.92·23-s − 5.00i·25-s + (−20.3 − 20.3i)27-s + (−38.8 − 38.8i)29-s + 2.05i·31-s + ⋯
L(s)  = 1  + (−0.641 + 0.641i)3-s + (0.316 − 0.316i)5-s − 0.587·7-s + 0.175i·9-s + (0.0307 + 0.0307i)11-s + (−0.912 − 0.912i)13-s + 0.406i·15-s − 0.539·17-s + (0.273 − 0.273i)19-s + (0.376 − 0.376i)21-s − 0.257·23-s − 0.200i·25-s + (−0.754 − 0.754i)27-s + (−1.34 − 1.34i)29-s + 0.0663i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.612 + 0.790i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -0.612 + 0.790i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.132758 - 0.270817i\)
\(L(\frac12)\) \(\approx\) \(0.132758 - 0.270817i\)
\(L(2)\) 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 \( 1 + (-1.58 + 1.58i)T \)
good3 \( 1 + (1.92 - 1.92i)T - 9iT^{2} \)
7 \( 1 + 4.10T + 49T^{2} \)
11 \( 1 + (-0.338 - 0.338i)T + 121iT^{2} \)
13 \( 1 + (11.8 + 11.8i)T + 169iT^{2} \)
17 \( 1 + 9.17T + 289T^{2} \)
19 \( 1 + (-5.20 + 5.20i)T - 361iT^{2} \)
23 \( 1 + 5.92T + 529T^{2} \)
29 \( 1 + (38.8 + 38.8i)T + 841iT^{2} \)
31 \( 1 - 2.05iT - 961T^{2} \)
37 \( 1 + (-38.1 + 38.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 40.1iT - 1.68e3T^{2} \)
43 \( 1 + (59.8 + 59.8i)T + 1.84e3iT^{2} \)
47 \( 1 + 57.4iT - 2.20e3T^{2} \)
53 \( 1 + (16.7 - 16.7i)T - 2.80e3iT^{2} \)
59 \( 1 + (9.95 + 9.95i)T + 3.48e3iT^{2} \)
61 \( 1 + (-65.8 - 65.8i)T + 3.72e3iT^{2} \)
67 \( 1 + (85.6 - 85.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 70.3T + 5.04e3T^{2} \)
73 \( 1 - 73.9iT - 5.32e3T^{2} \)
79 \( 1 - 114. iT - 6.24e3T^{2} \)
83 \( 1 + (24.0 - 24.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 13.2iT - 7.92e3T^{2} \)
97 \( 1 - 7.27T + 9.40e3T^{2} \)
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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

−11.04535096563133467197436922482, −10.00102806982858143645071106275, −9.616935238115528506405895402906, −8.276549201373495139286397140790, −7.16571314243246185309558862132, −5.86570633099614938324443189614, −5.15898463378100665129170823960, −4.01998252655733200728705289248, −2.41390942156073696510374959458, −0.14652158117014477990525028175, 1.74896301013098419510387660558, 3.32098792358788357873915911986, 4.86539305438189864549803959214, 6.15143025416752835241353846585, 6.72882715059662281647591041968, 7.64709727466585125421737131471, 9.184007121468554592911386363885, 9.774962339937803320920924181976, 11.02513231427168722071305868820, 11.76048701030961548524444367091

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