Properties

Degree 2
Conductor 5
Sign $0.0242 - 0.999i$
Motivic weight 17
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 660. i·2-s − 1.13e4i·3-s − 3.05e5·4-s + (−2.11e4 + 8.73e5i)5-s − 7.47e6·6-s − 2.37e7i·7-s + 1.15e8i·8-s + 1.24e6·9-s + (5.76e8 + 1.39e7i)10-s − 3.40e8·11-s + 3.45e9i·12-s + 6.20e8i·13-s − 1.56e10·14-s + (9.87e9 + 2.39e8i)15-s + 3.59e10·16-s − 9.78e9i·17-s + ⋯
L(s)  = 1  − 1.82i·2-s − 0.995i·3-s − 2.32·4-s + (−0.0242 + 0.999i)5-s − 1.81·6-s − 1.55i·7-s + 2.42i·8-s + 0.00960·9-s + (1.82 + 0.0442i)10-s − 0.478·11-s + 2.31i·12-s + 0.210i·13-s − 2.83·14-s + (0.994 + 0.0241i)15-s + 2.09·16-s − 0.340i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0242 - 0.999i)\, \overline{\Lambda}(18-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.0242 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $0.0242 - 0.999i$
motivic weight  =  \(17\)
character  :  $\chi_{5} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 5,\ (\ :17/2),\ 0.0242 - 0.999i)$
$L(9)$  $\approx$  $0.661618 + 0.645759i$
$L(\frac12)$  $\approx$  $0.661618 + 0.645759i$
$L(\frac{19}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5$, \(F_p\) is a polynomial of degree 2. If $p = 5$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + (2.11e4 - 8.73e5i)T \)
good2 \( 1 + 660. iT - 1.31e5T^{2} \)
3 \( 1 + 1.13e4iT - 1.29e8T^{2} \)
7 \( 1 + 2.37e7iT - 2.32e14T^{2} \)
11 \( 1 + 3.40e8T + 5.05e17T^{2} \)
13 \( 1 - 6.20e8iT - 8.65e18T^{2} \)
17 \( 1 + 9.78e9iT - 8.27e20T^{2} \)
19 \( 1 + 3.92e10T + 5.48e21T^{2} \)
23 \( 1 - 2.44e11iT - 1.41e23T^{2} \)
29 \( 1 + 3.98e12T + 7.25e24T^{2} \)
31 \( 1 - 2.84e12T + 2.25e25T^{2} \)
37 \( 1 + 2.87e13iT - 4.56e26T^{2} \)
41 \( 1 - 2.58e13T + 2.61e27T^{2} \)
43 \( 1 + 7.92e13iT - 5.87e27T^{2} \)
47 \( 1 + 3.42e13iT - 2.66e28T^{2} \)
53 \( 1 + 6.28e14iT - 2.05e29T^{2} \)
59 \( 1 - 1.43e15T + 1.27e30T^{2} \)
61 \( 1 + 1.41e15T + 2.24e30T^{2} \)
67 \( 1 - 3.52e14iT - 1.10e31T^{2} \)
71 \( 1 - 5.76e15T + 2.96e31T^{2} \)
73 \( 1 - 1.24e15iT - 4.74e31T^{2} \)
79 \( 1 + 8.16e15T + 1.81e32T^{2} \)
83 \( 1 + 2.09e16iT - 4.21e32T^{2} \)
89 \( 1 + 2.91e16T + 1.37e33T^{2} \)
97 \( 1 - 4.58e16iT - 5.95e33T^{2} \)
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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

−18.76380678809428641002948778052, −17.63895411100998515324843931022, −13.97059121943516090253795898691, −13.01325077663794311172844152039, −11.26736725668730067408136636297, −10.12107159547421820130246946691, −7.35592137097598665093700409895, −3.80901103716293605637354387970, −2.02167465799960000664627427316, −0.45987989328136317256592664141, 4.61312861597637042206629989323, 5.75839417192414805430177023377, 8.321300713309252373962981240855, 9.420505479614196650603761654917, 12.84509780401604970180215540129, 15.03388550416078894817418428872, 15.74259179206354578381779871500, 16.85574370268143149970430738791, 18.50257497010638558229581660427, 21.16297328605092292208067085614

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