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Properties

Label 2-320-16.3-c2-0-3
Degree $2$
Conductor $320$
Sign $-0.509 - 0.860i$
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  + (−3.09 + 3.09i)3-s + (−1.58 + 1.58i)5-s + 13.0·7-s − 10.2i·9-s + (4.91 + 4.91i)11-s + (9.80 + 9.80i)13-s − 9.79i·15-s − 0.0570·17-s + (−6.54 + 6.54i)19-s + (−40.3 + 40.3i)21-s − 12.8·23-s − 5.00i·25-s + (3.72 + 3.72i)27-s + (−20.3 − 20.3i)29-s + 60.4i·31-s + ⋯
L(s)  = 1  + (−1.03 + 1.03i)3-s + (−0.316 + 0.316i)5-s + 1.86·7-s − 1.13i·9-s + (0.446 + 0.446i)11-s + (0.753 + 0.753i)13-s − 0.653i·15-s − 0.00335·17-s + (−0.344 + 0.344i)19-s + (−1.92 + 1.92i)21-s − 0.560·23-s − 0.200i·25-s + (0.137 + 0.137i)27-s + (−0.702 − 0.702i)29-s + 1.94i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.509 - 0.860i$
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: \((2,\ 320,\ (\ :1),\ -0.509 - 0.860i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.611328 + 1.07284i\)
\(L(\frac12)\) \(\approx\) \(0.611328 + 1.07284i\)
\(L(2)\) 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 \( 1 + (3.09 - 3.09i)T - 9iT^{2} \)
7 \( 1 - 13.0T + 49T^{2} \)
11 \( 1 + (-4.91 - 4.91i)T + 121iT^{2} \)
13 \( 1 + (-9.80 - 9.80i)T + 169iT^{2} \)
17 \( 1 + 0.0570T + 289T^{2} \)
19 \( 1 + (6.54 - 6.54i)T - 361iT^{2} \)
23 \( 1 + 12.8T + 529T^{2} \)
29 \( 1 + (20.3 + 20.3i)T + 841iT^{2} \)
31 \( 1 - 60.4iT - 961T^{2} \)
37 \( 1 + (19.7 - 19.7i)T - 1.36e3iT^{2} \)
41 \( 1 - 33.6iT - 1.68e3T^{2} \)
43 \( 1 + (-16.9 - 16.9i)T + 1.84e3iT^{2} \)
47 \( 1 + 67.8iT - 2.20e3T^{2} \)
53 \( 1 + (-8.59 + 8.59i)T - 2.80e3iT^{2} \)
59 \( 1 + (-30.7 - 30.7i)T + 3.48e3iT^{2} \)
61 \( 1 + (-11.8 - 11.8i)T + 3.72e3iT^{2} \)
67 \( 1 + (18.9 - 18.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 110.T + 5.04e3T^{2} \)
73 \( 1 + 50.4iT - 5.32e3T^{2} \)
79 \( 1 - 77.2iT - 6.24e3T^{2} \)
83 \( 1 + (-109. + 109. i)T - 6.88e3iT^{2} \)
89 \( 1 - 93.1iT - 7.92e3T^{2} \)
97 \( 1 + 25.2T + 9.40e3T^{2} \)
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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

−11.65867489014511180929145151847, −10.85114036128293561797506925069, −10.23665601022065835321655535014, −8.944930538299177587844224863004, −8.015204452757648590101398831639, −6.75686080150916361776343515809, −5.56274836774496092053093852950, −4.62697613780256053738192890942, −3.94684724844997925034433957892, −1.64628684311656428968279713354, 0.73427318046557837434720894143, 1.84145429026692857034953265436, 4.10038541034246244644400198315, 5.33441942802032583841720619599, 6.02599133167014944981108018875, 7.37677679981226139361280070763, 8.016175336598734487791840026318, 8.935304079241054648228733030715, 10.75435881287857950168532399144, 11.24091679744797256037564979360

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