Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $0.258 - 0.965i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 30.0i·3-s + 25i·5-s + 45.4·7-s − 658.·9-s − 142. i·11-s − 897. i·13-s + 750.·15-s + 11.4·17-s + 1.81e3i·19-s − 1.36e3i·21-s − 3.87e3·23-s − 625·25-s + 1.24e4i·27-s + 3.46e3i·29-s − 1.55e3·31-s + ⋯
L(s)  = 1  − 1.92i·3-s + 0.447i·5-s + 0.350·7-s − 2.70·9-s − 0.355i·11-s − 1.47i·13-s + 0.861·15-s + 0.00964·17-s + 1.15i·19-s − 0.675i·21-s − 1.52·23-s − 0.200·25-s + 3.29i·27-s + 0.764i·29-s − 0.289·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.258 - 0.965i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (161, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.258 - 0.965i)\)
\(L(3)\)  \(\approx\)  \(0.01293959086\)
\(L(\frac12)\)  \(\approx\)  \(0.01293959086\)
\(L(\frac{7}{2})\)   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 \( 1 - 25iT \)
good3 \( 1 + 30.0iT - 243T^{2} \)
7 \( 1 - 45.4T + 1.68e4T^{2} \)
11 \( 1 + 142. iT - 1.61e5T^{2} \)
13 \( 1 + 897. iT - 3.71e5T^{2} \)
17 \( 1 - 11.4T + 1.41e6T^{2} \)
19 \( 1 - 1.81e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.87e3T + 6.43e6T^{2} \)
29 \( 1 - 3.46e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.55e3T + 2.86e7T^{2} \)
37 \( 1 + 9.61e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.48e4T + 1.15e8T^{2} \)
43 \( 1 - 6.32e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.90e4T + 2.29e8T^{2} \)
53 \( 1 - 3.25e4iT - 4.18e8T^{2} \)
59 \( 1 - 652. iT - 7.14e8T^{2} \)
61 \( 1 + 4.71e3iT - 8.44e8T^{2} \)
67 \( 1 - 4.63e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.26e4T + 1.80e9T^{2} \)
73 \( 1 - 6.93e4T + 2.07e9T^{2} \)
79 \( 1 - 3.32e4T + 3.07e9T^{2} \)
83 \( 1 + 5.53e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.32e4T + 5.58e9T^{2} \)
97 \( 1 + 8.24e4T + 8.58e9T^{2} \)
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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

−11.17244959286391372524899336482, −10.29157849599949515085088365548, −8.659973672268637232543389486481, −7.891493289273937647390027820452, −7.34771865180356783934703693605, −6.11863537985661730747237568595, −5.58858850171765886294077236087, −3.42427010304379260191520080691, −2.29574393430344780455877861510, −1.21640516625785591145220218716, 0.00341728204513170318580187366, 2.19128296732849287556235814977, 3.74144792584067199636818288793, 4.50594969953705116820627788118, 5.20735171618854972500427423300, 6.48624158207924985536059089194, 8.138741969070805758060256893351, 9.020383495485476660524722922567, 9.636393514623307363853619384047, 10.44790606515421502116062210230

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