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Properties

Label 2-320-5.2-c2-0-0
Degree $2$
Conductor $320$
Sign $-0.850 + 0.525i$
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  + (−2 + 2i)3-s + 5i·5-s + (−2 − 2i)7-s + i·9-s − 8·11-s + (−3 + 3i)13-s + (−10 − 10i)15-s + (7 + 7i)17-s − 20i·19-s + 8·21-s + (2 − 2i)23-s − 25·25-s + (−20 − 20i)27-s − 40i·29-s − 52·31-s + ⋯
L(s)  = 1  + (−0.666 + 0.666i)3-s + i·5-s + (−0.285 − 0.285i)7-s + 0.111i·9-s − 0.727·11-s + (−0.230 + 0.230i)13-s + (−0.666 − 0.666i)15-s + (0.411 + 0.411i)17-s − 1.05i·19-s + 0.380·21-s + (0.0869 − 0.0869i)23-s − 25-s + (−0.740 − 0.740i)27-s − 1.37i·29-s − 1.67·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0840686 - 0.295933i\)
\(L(\frac12)\) \(\approx\) \(0.0840686 - 0.295933i\)
\(L(2)\) 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 \( 1 - 5iT \)
good3 \( 1 + (2 - 2i)T - 9iT^{2} \)
7 \( 1 + (2 + 2i)T + 49iT^{2} \)
11 \( 1 + 8T + 121T^{2} \)
13 \( 1 + (3 - 3i)T - 169iT^{2} \)
17 \( 1 + (-7 - 7i)T + 289iT^{2} \)
19 \( 1 + 20iT - 361T^{2} \)
23 \( 1 + (-2 + 2i)T - 529iT^{2} \)
29 \( 1 + 40iT - 841T^{2} \)
31 \( 1 + 52T + 961T^{2} \)
37 \( 1 + (-3 - 3i)T + 1.36e3iT^{2} \)
41 \( 1 + 8T + 1.68e3T^{2} \)
43 \( 1 + (42 - 42i)T - 1.84e3iT^{2} \)
47 \( 1 + (-18 - 18i)T + 2.20e3iT^{2} \)
53 \( 1 + (53 - 53i)T - 2.80e3iT^{2} \)
59 \( 1 + 20iT - 3.48e3T^{2} \)
61 \( 1 - 48T + 3.72e3T^{2} \)
67 \( 1 + (-62 - 62i)T + 4.48e3iT^{2} \)
71 \( 1 - 28T + 5.04e3T^{2} \)
73 \( 1 + (47 - 47i)T - 5.32e3iT^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + (-18 + 18i)T - 6.88e3iT^{2} \)
89 \( 1 - 80iT - 7.92e3T^{2} \)
97 \( 1 + (63 + 63i)T + 9.40e3iT^{2} \)
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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

−11.55194846689576645003998743273, −10.99834316298873859013557737102, −10.22860599889668295577539666552, −9.554768596656884400193559313600, −8.018386157490094493965855948608, −7.09242078286641234963548964427, −6.03038595405425844633404003888, −5.01221489230905050568642566682, −3.81104158227833622850184445117, −2.46081232413345541203920320615, 0.15370778222260220337276272857, 1.64546213545104287536603378777, 3.51031135718440197638193407300, 5.17232278367607381913452762478, 5.71196067385763554615819402934, 6.97435986619306822706848867055, 7.929148883339192602668817054321, 8.979484677752577776555249991258, 9.888361864340952534128036702616, 11.03301340543752415345463926368

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