Properties

Degree 2
Conductor $ 2^{4} \cdot 5 \cdot 7 $
Sign $-0.899 - 0.437i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.58 − 1.58i)3-s + (−1.58 − 1.58i)5-s + (0.581 − 2.58i)7-s + 2.00i·9-s + 11-s + (−1.58 − 1.58i)13-s + 5.00i·15-s + (1.58 − 1.58i)17-s − 3.16·19-s + (−5 + 3.16i)21-s + (−2 + 2i)23-s + 5.00i·25-s + (−1.58 + 1.58i)27-s + 3i·29-s + 3.16i·31-s + ⋯
L(s)  = 1  + (−0.912 − 0.912i)3-s + (−0.707 − 0.707i)5-s + (0.219 − 0.975i)7-s + 0.666i·9-s + 0.301·11-s + (−0.438 − 0.438i)13-s + 1.29i·15-s + (0.383 − 0.383i)17-s − 0.725·19-s + (−1.09 + 0.690i)21-s + (−0.417 + 0.417i)23-s + 1.00i·25-s + (−0.304 + 0.304i)27-s + 0.557i·29-s + 0.567i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.899 - 0.437i$
motivic weight  =  \(1\)
character  :  $\chi_{560} (97, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 560,\ (\ :1/2),\ -0.899 - 0.437i)\)
\(L(1)\)  \(\approx\)  \(0.109684 + 0.475645i\)
\(L(\frac12)\)  \(\approx\)  \(0.109684 + 0.475645i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.58 + 1.58i)T \)
7 \( 1 + (-0.581 + 2.58i)T \)
good3 \( 1 + (1.58 + 1.58i)T + 3iT^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (1.58 + 1.58i)T + 13iT^{2} \)
17 \( 1 + (-1.58 + 1.58i)T - 17iT^{2} \)
19 \( 1 + 3.16T + 19T^{2} \)
23 \( 1 + (2 - 2i)T - 23iT^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 - 3.16iT - 31T^{2} \)
37 \( 1 + (6 + 6i)T + 37iT^{2} \)
41 \( 1 - 9.48iT - 41T^{2} \)
43 \( 1 + (-3 + 3i)T - 43iT^{2} \)
47 \( 1 + (-4.74 + 4.74i)T - 47iT^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 + 9.48T + 59T^{2} \)
61 \( 1 + 6.32iT - 61T^{2} \)
67 \( 1 + (-1 - i)T + 67iT^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 13iT - 79T^{2} \)
83 \( 1 + (-3.16 - 3.16i)T + 83iT^{2} \)
89 \( 1 + 6.32T + 89T^{2} \)
97 \( 1 + (1.58 - 1.58i)T - 97iT^{2} \)
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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

−10.54758968347775904254801010385, −9.368264995425823716117131779072, −8.220858559129275215423798789154, −7.43275282023006510434238821749, −6.80956830735188032586319270898, −5.61972507577436832496684517610, −4.70202886114552999682199541047, −3.58411013986060227668253818799, −1.49650498188424158297971662093, −0.32324884696239103516372347832, 2.39107749742439765669117329982, 3.85688045322793135192724766793, 4.66546324608245500627467459546, 5.75380949491179843479477291476, 6.51310654736435271553296441340, 7.72870501283941326180974429356, 8.694828188734283794657853881865, 9.725213811817024364816323714274, 10.54455167449710961359205448513, 11.19232534297574396532684252562

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