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Properties

Label 2-320-16.11-c2-0-7
Degree $2$
Conductor $320$
Sign $0.961 + 0.273i$
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  + (−0.0991 − 0.0991i)3-s + (1.58 + 1.58i)5-s + 3.75·7-s − 8.98i·9-s + (1.41 − 1.41i)11-s + (3.89 − 3.89i)13-s − 0.313i·15-s − 5.09·17-s + (20.4 + 20.4i)19-s + (−0.371 − 0.371i)21-s + 24.0·23-s + 5.00i·25-s + (−1.78 + 1.78i)27-s + (32.0 − 32.0i)29-s − 26.0i·31-s + ⋯
L(s)  = 1  + (−0.0330 − 0.0330i)3-s + (0.316 + 0.316i)5-s + 0.536·7-s − 0.997i·9-s + (0.128 − 0.128i)11-s + (0.299 − 0.299i)13-s − 0.0208i·15-s − 0.299·17-s + (1.07 + 1.07i)19-s + (−0.0177 − 0.0177i)21-s + 1.04·23-s + 0.200i·25-s + (−0.0660 + 0.0660i)27-s + (1.10 − 1.10i)29-s − 0.840i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.961 + 0.273i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ 0.961 + 0.273i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.86065 - 0.259605i\)
\(L(\frac12)\) \(\approx\) \(1.86065 - 0.259605i\)
\(L(2)\) 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 \( 1 + (0.0991 + 0.0991i)T + 9iT^{2} \)
7 \( 1 - 3.75T + 49T^{2} \)
11 \( 1 + (-1.41 + 1.41i)T - 121iT^{2} \)
13 \( 1 + (-3.89 + 3.89i)T - 169iT^{2} \)
17 \( 1 + 5.09T + 289T^{2} \)
19 \( 1 + (-20.4 - 20.4i)T + 361iT^{2} \)
23 \( 1 - 24.0T + 529T^{2} \)
29 \( 1 + (-32.0 + 32.0i)T - 841iT^{2} \)
31 \( 1 + 26.0iT - 961T^{2} \)
37 \( 1 + (-25.9 - 25.9i)T + 1.36e3iT^{2} \)
41 \( 1 + 3.03iT - 1.68e3T^{2} \)
43 \( 1 + (-9.98 + 9.98i)T - 1.84e3iT^{2} \)
47 \( 1 + 73.9iT - 2.20e3T^{2} \)
53 \( 1 + (-58.6 - 58.6i)T + 2.80e3iT^{2} \)
59 \( 1 + (64.2 - 64.2i)T - 3.48e3iT^{2} \)
61 \( 1 + (9.51 - 9.51i)T - 3.72e3iT^{2} \)
67 \( 1 + (11.3 + 11.3i)T + 4.48e3iT^{2} \)
71 \( 1 + 74.9T + 5.04e3T^{2} \)
73 \( 1 - 81.1iT - 5.32e3T^{2} \)
79 \( 1 - 8.43iT - 6.24e3T^{2} \)
83 \( 1 + (-53.4 - 53.4i)T + 6.88e3iT^{2} \)
89 \( 1 + 116. iT - 7.92e3T^{2} \)
97 \( 1 + 148.T + 9.40e3T^{2} \)
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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

−11.50734639935869231821285457360, −10.39591158952023712146000740641, −9.565416412718368960782124266968, −8.574231885987534361390886593730, −7.53080455102331759549396773034, −6.42184895517277334130055732987, −5.55132038582245042233763893103, −4.13503655129746700793083100798, −2.88860891936429801738169865516, −1.13448344010155369478042710227, 1.36976940377324469826705545814, 2.85496547671491525181926078373, 4.63709808321557627760067518027, 5.22752717463742043056420710475, 6.65827173645131801198279600139, 7.65809964010269731100135386171, 8.699170565484769491244461327936, 9.483159726627043580249370765378, 10.74213240072850418518254986216, 11.25256614432630608073940896952

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