Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.755·3-s + 25·5-s − 172.·7-s − 242.·9-s − 391.·11-s − 149.·13-s + 18.8·15-s + 1.18e3·17-s − 685.·19-s − 130.·21-s + 996.·23-s + 625·25-s − 366.·27-s + 8.76e3·29-s + 9.52e3·31-s − 295.·33-s − 4.31e3·35-s − 1.02e4·37-s − 112.·39-s + 32.6·41-s − 1.03e4·43-s − 6.06e3·45-s + 1.69e4·47-s + 1.29e4·49-s + 897.·51-s + 2.22e4·53-s − 9.78e3·55-s + ⋯
L(s)  = 1  + 0.0484·3-s + 0.447·5-s − 1.33·7-s − 0.997·9-s − 0.975·11-s − 0.244·13-s + 0.0216·15-s + 0.996·17-s − 0.435·19-s − 0.0644·21-s + 0.392·23-s + 0.200·25-s − 0.0968·27-s + 1.93·29-s + 1.78·31-s − 0.0472·33-s − 0.594·35-s − 1.22·37-s − 0.0118·39-s + 0.00303·41-s − 0.851·43-s − 0.446·45-s + 1.11·47-s + 0.768·49-s + 0.0483·51-s + 1.08·53-s − 0.436·55-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{320} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.374828783\)
\(L(\frac12)\)  \(\approx\)  \(1.374828783\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - 25T \)
good3 \( 1 - 0.755T + 243T^{2} \)
7 \( 1 + 172.T + 1.68e4T^{2} \)
11 \( 1 + 391.T + 1.61e5T^{2} \)
13 \( 1 + 149.T + 3.71e5T^{2} \)
17 \( 1 - 1.18e3T + 1.41e6T^{2} \)
19 \( 1 + 685.T + 2.47e6T^{2} \)
23 \( 1 - 996.T + 6.43e6T^{2} \)
29 \( 1 - 8.76e3T + 2.05e7T^{2} \)
31 \( 1 - 9.52e3T + 2.86e7T^{2} \)
37 \( 1 + 1.02e4T + 6.93e7T^{2} \)
41 \( 1 - 32.6T + 1.15e8T^{2} \)
43 \( 1 + 1.03e4T + 1.47e8T^{2} \)
47 \( 1 - 1.69e4T + 2.29e8T^{2} \)
53 \( 1 - 2.22e4T + 4.18e8T^{2} \)
59 \( 1 + 4.42e4T + 7.14e8T^{2} \)
61 \( 1 - 2.17e4T + 8.44e8T^{2} \)
67 \( 1 + 2.07e4T + 1.35e9T^{2} \)
71 \( 1 - 1.53e4T + 1.80e9T^{2} \)
73 \( 1 - 5.79e4T + 2.07e9T^{2} \)
79 \( 1 + 6.24e4T + 3.07e9T^{2} \)
83 \( 1 - 4.35e4T + 3.93e9T^{2} \)
89 \( 1 - 6.62e4T + 5.58e9T^{2} \)
97 \( 1 + 1.22e4T + 8.58e9T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.44670452373138829520860713213, −10.06817175259641710840199890286, −8.931408507270795468616980361898, −8.064791440586168670673700734149, −6.75459541166765416233044427158, −5.94694881407921454603945652696, −4.94500745244073658642502318583, −3.23825326746480008611560361120, −2.57211770922465627480113105286, −0.62161560557020551595382691199, 0.62161560557020551595382691199, 2.57211770922465627480113105286, 3.23825326746480008611560361120, 4.94500745244073658642502318583, 5.94694881407921454603945652696, 6.75459541166765416233044427158, 8.064791440586168670673700734149, 8.931408507270795468616980361898, 10.06817175259641710840199890286, 10.44670452373138829520860713213

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