Properties

Degree 2
Conductor 5
Sign $-0.396 - 0.917i$
Motivic weight 17
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 512. i·2-s − 1.82e4i·3-s − 1.31e5·4-s + (3.46e5 + 8.01e5i)5-s + 9.35e6·6-s + 1.34e7i·7-s − 3.36e5i·8-s − 2.03e8·9-s + (−4.11e8 + 1.77e8i)10-s + 8.20e8·11-s + 2.40e9i·12-s + 2.92e9i·13-s − 6.91e9·14-s + (1.46e10 − 6.32e9i)15-s − 1.70e10·16-s + 3.73e10i·17-s + ⋯
L(s)  = 1  + 1.41i·2-s − 1.60i·3-s − 1.00·4-s + (0.396 + 0.917i)5-s + 2.27·6-s + 0.883i·7-s − 0.00708i·8-s − 1.57·9-s + (−1.29 + 0.561i)10-s + 1.15·11-s + 1.61i·12-s + 0.995i·13-s − 1.25·14-s + (1.47 − 0.637i)15-s − 0.994·16-s + 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.917i)\, \overline{\Lambda}(18-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $-0.396 - 0.917i$
motivic weight  =  \(17\)
character  :  $\chi_{5} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 5,\ (\ :17/2),\ -0.396 - 0.917i)$
$L(9)$  $\approx$  $0.934291 + 1.42184i$
$L(\frac12)$  $\approx$  $0.934291 + 1.42184i$
$L(\frac{19}{2})$   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 \( 1 + (-3.46e5 - 8.01e5i)T \)
good2 \( 1 - 512. iT - 1.31e5T^{2} \)
3 \( 1 + 1.82e4iT - 1.29e8T^{2} \)
7 \( 1 - 1.34e7iT - 2.32e14T^{2} \)
11 \( 1 - 8.20e8T + 5.05e17T^{2} \)
13 \( 1 - 2.92e9iT - 8.65e18T^{2} \)
17 \( 1 - 3.73e10iT - 8.27e20T^{2} \)
19 \( 1 - 3.00e10T + 5.48e21T^{2} \)
23 \( 1 + 4.40e11iT - 1.41e23T^{2} \)
29 \( 1 - 1.80e12T + 7.25e24T^{2} \)
31 \( 1 - 2.90e12T + 2.25e25T^{2} \)
37 \( 1 + 3.31e12iT - 4.56e26T^{2} \)
41 \( 1 + 7.87e13T + 2.61e27T^{2} \)
43 \( 1 + 4.61e12iT - 5.87e27T^{2} \)
47 \( 1 + 1.39e14iT - 2.66e28T^{2} \)
53 \( 1 + 4.39e13iT - 2.05e29T^{2} \)
59 \( 1 + 3.18e14T + 1.27e30T^{2} \)
61 \( 1 - 7.17e14T + 2.24e30T^{2} \)
67 \( 1 - 2.87e15iT - 1.10e31T^{2} \)
71 \( 1 + 3.39e14T + 2.96e31T^{2} \)
73 \( 1 + 8.49e15iT - 4.74e31T^{2} \)
79 \( 1 - 1.05e16T + 1.81e32T^{2} \)
83 \( 1 - 2.86e16iT - 4.21e32T^{2} \)
89 \( 1 - 6.62e16T + 1.37e33T^{2} \)
97 \( 1 + 2.94e16iT - 5.95e33T^{2} \)
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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

−19.21668909675786481903118149209, −18.18968761962573107769955028632, −17.02959175418687169427174014080, −14.86236295140120010930588680825, −13.85123753016774665154034295070, −11.90792624302271829063243322375, −8.548526298404405942343326454875, −6.83204213793252780556606918871, −6.17580734080903143768899000099, −2.00376158438617532957011914310, 0.894398863380974756150534442666, 3.44869972202378375392138932058, 4.77557233054311840163398467213, 9.296866998457131771775053845251, 10.19603144294750652297749115150, 11.71392849450218946826649132779, 13.70823142126381458813132845778, 15.90828146563584397004803387383, 17.30497796478639977573718878478, 19.98659418950675523915850896150

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