Properties

Degree $2$
Conductor $320$
Sign $-0.994 + 0.106i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 19.1·3-s + (55.2 + 8.66i)5-s + 121. i·7-s + 123.·9-s + 298i·11-s − 485.·13-s + (−1.05e3 − 165. i)15-s − 420. i·17-s + 2.57e3i·19-s − 2.32e3i·21-s + 717. i·23-s + (2.97e3 + 956. i)25-s + 2.29e3·27-s − 3.73e3i·29-s + 6.22e3·31-s + ⋯
L(s)  = 1  − 1.22·3-s + (0.987 + 0.154i)5-s + 0.937i·7-s + 0.506·9-s + 0.742i·11-s − 0.797·13-s + (−1.21 − 0.190i)15-s − 0.353i·17-s + 1.63i·19-s − 1.15i·21-s + 0.282i·23-s + (0.952 + 0.306i)25-s + 0.606·27-s − 0.824i·29-s + 1.16·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.994 + 0.106i$
Motivic weight: \(5\)
Character: $\chi_{320} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ -0.994 + 0.106i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5366413978\)
\(L(\frac12)\) \(\approx\) \(0.5366413978\)
\(L(\frac{7}{2})\) 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 \( 1 + (-55.2 - 8.66i)T \)
good3 \( 1 + 19.1T + 243T^{2} \)
7 \( 1 - 121. iT - 1.68e4T^{2} \)
11 \( 1 - 298iT - 1.61e5T^{2} \)
13 \( 1 + 485.T + 3.71e5T^{2} \)
17 \( 1 + 420. iT - 1.41e6T^{2} \)
19 \( 1 - 2.57e3iT - 2.47e6T^{2} \)
23 \( 1 - 717. iT - 6.43e6T^{2} \)
29 \( 1 + 3.73e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.22e3T + 2.86e7T^{2} \)
37 \( 1 + 5.72e3T + 6.93e7T^{2} \)
41 \( 1 - 1.13e4T + 1.15e8T^{2} \)
43 \( 1 + 1.91e4T + 1.47e8T^{2} \)
47 \( 1 + 2.28e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.72e4T + 4.18e8T^{2} \)
59 \( 1 + 1.63e4iT - 7.14e8T^{2} \)
61 \( 1 - 4.87e4iT - 8.44e8T^{2} \)
67 \( 1 + 8.39e3T + 1.35e9T^{2} \)
71 \( 1 + 2.90e4T + 1.80e9T^{2} \)
73 \( 1 + 420. iT - 2.07e9T^{2} \)
79 \( 1 + 3.11e4T + 3.07e9T^{2} \)
83 \( 1 + 1.06e5T + 3.93e9T^{2} \)
89 \( 1 + 9.93e4T + 5.58e9T^{2} \)
97 \( 1 + 1.80e5iT - 8.58e9T^{2} \)
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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

−11.43549055057487243767374920476, −10.13611864452427091317818852721, −9.839256417750570459537711505991, −8.545736692859866687370256216162, −7.18167564476570847258834593362, −6.10824767473581513102015253841, −5.57298993660533495180672644757, −4.63627020526938312961526069356, −2.70465077125751782100970246009, −1.55158919650606585430643920607, 0.17910271212036626436635804063, 1.16831888350469217130965480641, 2.83314352951830629074152949656, 4.58483398711268216371129150238, 5.31776056834574124469597201183, 6.38061526438215291960934968734, 7.02437210273869480695895171902, 8.504693313299755175461550995563, 9.614231139222544214800991409476, 10.54348732682148822325023206317

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