Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-0.897 - 0.441i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.4·3-s + (−17.9 − 52.9i)5-s − 86.7i·7-s − 134.·9-s − 258. i·11-s + 465.·13-s + (187. + 551. i)15-s − 2.02e3i·17-s − 1.05e3i·19-s + 904. i·21-s − 3.99e3i·23-s + (−2.47e3 + 1.90e3i)25-s + 3.93e3·27-s + 3.24e3i·29-s − 991.·31-s + ⋯
L(s)  = 1  − 0.668·3-s + (−0.321 − 0.946i)5-s − 0.669i·7-s − 0.552·9-s − 0.644i·11-s + 0.763·13-s + (0.215 + 0.633i)15-s − 1.70i·17-s − 0.667i·19-s + 0.447i·21-s − 1.57i·23-s + (−0.792 + 0.609i)25-s + 1.03·27-s + 0.715i·29-s − 0.185·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.897 - 0.441i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -0.897 - 0.441i)\)
\(L(3)\)  \(\approx\)  \(0.7846100361\)
\(L(\frac12)\)  \(\approx\)  \(0.7846100361\)
\(L(\frac{7}{2})\)   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 \( 1 + (17.9 + 52.9i)T \)
good3 \( 1 + 10.4T + 243T^{2} \)
7 \( 1 + 86.7iT - 1.68e4T^{2} \)
11 \( 1 + 258. iT - 1.61e5T^{2} \)
13 \( 1 - 465.T + 3.71e5T^{2} \)
17 \( 1 + 2.02e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.05e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.99e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.24e3iT - 2.05e7T^{2} \)
31 \( 1 + 991.T + 2.86e7T^{2} \)
37 \( 1 - 8.82e3T + 6.93e7T^{2} \)
41 \( 1 - 5.63e3T + 1.15e8T^{2} \)
43 \( 1 + 1.77e3T + 1.47e8T^{2} \)
47 \( 1 + 1.77e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.59e4T + 4.18e8T^{2} \)
59 \( 1 + 1.81e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.54e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.20e4T + 1.35e9T^{2} \)
71 \( 1 + 7.13e4T + 1.80e9T^{2} \)
73 \( 1 - 3.25e3iT - 2.07e9T^{2} \)
79 \( 1 + 8.42e4T + 3.07e9T^{2} \)
83 \( 1 - 7.90e4T + 3.93e9T^{2} \)
89 \( 1 + 6.51e4T + 5.58e9T^{2} \)
97 \( 1 + 3.39e4iT - 8.58e9T^{2} \)
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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

−10.49967809074661552072135635624, −9.135936077357748798550820176995, −8.515414298180453661258432376705, −7.34005513710988213131386283563, −6.23023294179811914360033238378, −5.19084921205168924953673013439, −4.34645329932659395654533429422, −2.92814594923882754090090341511, −0.923662891989800893663854244573, −0.29025436164926788000856105119, 1.66953869193184611159839925655, 3.07330831542107247368761672870, 4.21185124354377366843858990596, 5.88562973865718196600251173310, 6.09351111345565389624270786137, 7.51782301946266690982836579908, 8.386833301533513510491185837834, 9.611819504955254579009270987559, 10.63725951453031447186108886870, 11.29468876547616043389009691836

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