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Properties

Label 2-320-16.13-c1-0-0
Degree $2$
Conductor $320$
Sign $0.802 - 0.596i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.82 − 1.82i)3-s + (−0.707 + 0.707i)5-s + 4.50i·7-s + 3.68i·9-s + (1.64 − 1.64i)11-s + (1.51 + 1.51i)13-s + 2.58·15-s + 1.45·17-s + (2.67 + 2.67i)19-s + (8.24 − 8.24i)21-s + 2.37i·23-s − 1.00i·25-s + (1.24 − 1.24i)27-s + (0.924 + 0.924i)29-s + 7.20·31-s + ⋯
L(s)  = 1  + (−1.05 − 1.05i)3-s + (−0.316 + 0.316i)5-s + 1.70i·7-s + 1.22i·9-s + (0.494 − 0.494i)11-s + (0.421 + 0.421i)13-s + 0.667·15-s + 0.353·17-s + (0.614 + 0.614i)19-s + (1.79 − 1.79i)21-s + 0.495i·23-s − 0.200i·25-s + (0.239 − 0.239i)27-s + (0.171 + 0.171i)29-s + 1.29·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.802 - 0.596i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.802 - 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.782562 + 0.258700i\)
\(L(\frac12)\) \(\approx\) \(0.782562 + 0.258700i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (1.82 + 1.82i)T + 3iT^{2} \)
7 \( 1 - 4.50iT - 7T^{2} \)
11 \( 1 + (-1.64 + 1.64i)T - 11iT^{2} \)
13 \( 1 + (-1.51 - 1.51i)T + 13iT^{2} \)
17 \( 1 - 1.45T + 17T^{2} \)
19 \( 1 + (-2.67 - 2.67i)T + 19iT^{2} \)
23 \( 1 - 2.37iT - 23T^{2} \)
29 \( 1 + (-0.924 - 0.924i)T + 29iT^{2} \)
31 \( 1 - 7.20T + 31T^{2} \)
37 \( 1 + (5.21 - 5.21i)T - 37iT^{2} \)
41 \( 1 - 6.41iT - 41T^{2} \)
43 \( 1 + (7.65 - 7.65i)T - 43iT^{2} \)
47 \( 1 - 2.51T + 47T^{2} \)
53 \( 1 + (-1.50 + 1.50i)T - 53iT^{2} \)
59 \( 1 + (-5.31 + 5.31i)T - 59iT^{2} \)
61 \( 1 + (1.02 + 1.02i)T + 61iT^{2} \)
67 \( 1 + (5.22 + 5.22i)T + 67iT^{2} \)
71 \( 1 + 1.92iT - 71T^{2} \)
73 \( 1 - 1.39iT - 73T^{2} \)
79 \( 1 + 5.06T + 79T^{2} \)
83 \( 1 + (-2.44 - 2.44i)T + 83iT^{2} \)
89 \( 1 + 9.36iT - 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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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

−11.71172315517903076713156378396, −11.39655398438151749218490339524, −9.930050969885592271306410018917, −8.737088699249460166958497436345, −7.86662047887083094400384534470, −6.56218749407795175855501776041, −6.05942768389190035610061317668, −5.08458029858295570157388670468, −3.15258777974236311583714727115, −1.53357315512767833986895715953, 0.74909416895864156142606767649, 3.68491556955770368355681040122, 4.40065229662738916320831859637, 5.32552306363575352672770828373, 6.64881078138693082832598768608, 7.55647576418831794117671529252, 8.913509433496339327336597777378, 10.20965755795330654867934254382, 10.37259312778410134648946872006, 11.42450216439130688724268164666

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