Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-0.811 - 0.583i$
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

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.811 - 0.583i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -0.811 - 0.583i)\)
\(L(3)\)  \(\approx\)  \(0.6508666191\)
\(L(\frac12)\)  \(\approx\)  \(0.6508666191\)
\(L(\frac{7}{2})\)   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

−11.49035813545491050950693019594, −10.29688312526893813993936443935, −9.100825521347212341775663145892, −8.752358659558085331948761904161, −7.08797929083153012272204895506, −5.99370519078308201669764170816, −5.30173835612859866038844654937, −4.62453571947518767698125936269, −2.54666246483995722290515690357, −1.26645769383061291103348130625, 0.23161568362197997364991343079, 1.28569105617803435336275914556, 3.32225813724803327983926701053, 4.23479289795953993089250908052, 5.95052610775695114507391503102, 6.16731158215985764268244009039, 7.32791637395907746698390700320, 8.347090988970929997700416963774, 10.07319072750605783631818811364, 10.58711330662739111592145695812

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