Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-0.583 + 0.811i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 19.3·3-s + (19.7 − 52.2i)5-s − 226. i·7-s + 129.·9-s − 533. i·11-s + 540.·13-s + (381. − 1.00e3i)15-s − 300. i·17-s + 2.82e3i·19-s − 4.37e3i·21-s − 1.51e3i·23-s + (−2.34e3 − 2.06e3i)25-s − 2.18e3·27-s + 5.35e3i·29-s − 2.54e3·31-s + ⋯
L(s)  = 1  + 1.23·3-s + (0.353 − 0.935i)5-s − 1.74i·7-s + 0.533·9-s − 1.32i·11-s + 0.887·13-s + (0.438 − 1.15i)15-s − 0.251i·17-s + 1.79i·19-s − 2.16i·21-s − 0.595i·23-s + (−0.749 − 0.661i)25-s − 0.577·27-s + 1.18i·29-s − 0.475·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 + 0.811i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.583 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.583 + 0.811i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -0.583 + 0.811i)\)
\(L(3)\)  \(\approx\)  \(3.164125781\)
\(L(\frac12)\)  \(\approx\)  \(3.164125781\)
\(L(\frac{7}{2})\)   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 \( 1 + (-19.7 + 52.2i)T \)
good3 \( 1 - 19.3T + 243T^{2} \)
7 \( 1 + 226. iT - 1.68e4T^{2} \)
11 \( 1 + 533. iT - 1.61e5T^{2} \)
13 \( 1 - 540.T + 3.71e5T^{2} \)
17 \( 1 + 300. iT - 1.41e6T^{2} \)
19 \( 1 - 2.82e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.51e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.35e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.54e3T + 2.86e7T^{2} \)
37 \( 1 + 2.94e3T + 6.93e7T^{2} \)
41 \( 1 + 1.97e3T + 1.15e8T^{2} \)
43 \( 1 + 9.17e3T + 1.47e8T^{2} \)
47 \( 1 - 8.14e3iT - 2.29e8T^{2} \)
53 \( 1 - 2.56e4T + 4.18e8T^{2} \)
59 \( 1 + 3.93e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.16e4iT - 8.44e8T^{2} \)
67 \( 1 - 6.96e4T + 1.35e9T^{2} \)
71 \( 1 - 7.48e4T + 1.80e9T^{2} \)
73 \( 1 - 5.74e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.21e4T + 3.07e9T^{2} \)
83 \( 1 - 2.65e3T + 3.93e9T^{2} \)
89 \( 1 - 1.44e4T + 5.58e9T^{2} \)
97 \( 1 - 9.72e4iT - 8.58e9T^{2} \)
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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

−10.34772191185386363684088880427, −9.419167021087027559982429967570, −8.327655563786116644221377656941, −8.103061497608988585548467140981, −6.70070840748186663529629583194, −5.44256021811528142111691775412, −3.93186760119193679173362582225, −3.43638156706339205261337210311, −1.64583445521617606048160898939, −0.67321490183100409947048229370, 2.07118833534026474155545963401, 2.49635975504523201875684378277, 3.62752679557947277906456965833, 5.22864177450272582490916013598, 6.35557563753080243218593507227, 7.37852863939163628828105070784, 8.497125863036526522184218418627, 9.190614401564067914748432823689, 9.860427752120404682607322180983, 11.17438369905657507206152087747

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