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Properties

Degree 2
Conductor 5
Sign $-0.710 + 0.703i$
Motivic weight 17
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 197. i·2-s − 1.28e4i·3-s + 9.21e4·4-s + (6.20e5 − 6.14e5i)5-s − 2.52e6·6-s + 4.49e6i·7-s − 4.40e7i·8-s − 3.51e7·9-s + (−1.21e8 − 1.22e8i)10-s − 1.08e9·11-s − 1.18e9i·12-s − 2.20e9i·13-s + 8.85e8·14-s + (−7.87e9 − 7.95e9i)15-s + 3.40e9·16-s + 4.96e10i·17-s + ⋯
L(s)  = 1  − 0.544i·2-s − 1.12i·3-s + 0.703·4-s + (0.710 − 0.703i)5-s − 0.614·6-s + 0.294i·7-s − 0.927i·8-s − 0.272·9-s + (−0.383 − 0.387i)10-s − 1.53·11-s − 0.793i·12-s − 0.749i·13-s + 0.160·14-s + (−0.793 − 0.801i)15-s + 0.198·16-s + 1.72i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(18-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $-0.710 + 0.703i$
motivic weight  =  \(17\)
character  :  $\chi_{5} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 5,\ (\ :17/2),\ -0.710 + 0.703i)$
$L(9)$  $\approx$  $0.811435 - 1.97336i$
$L(\frac12)$  $\approx$  $0.811435 - 1.97336i$
$L(\frac{19}{2})$   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 \( 1 + (-6.20e5 + 6.14e5i)T \)
good2 \( 1 + 197. iT - 1.31e5T^{2} \)
3 \( 1 + 1.28e4iT - 1.29e8T^{2} \)
7 \( 1 - 4.49e6iT - 2.32e14T^{2} \)
11 \( 1 + 1.08e9T + 5.05e17T^{2} \)
13 \( 1 + 2.20e9iT - 8.65e18T^{2} \)
17 \( 1 - 4.96e10iT - 8.27e20T^{2} \)
19 \( 1 - 6.53e10T + 5.48e21T^{2} \)
23 \( 1 - 6.03e10iT - 1.41e23T^{2} \)
29 \( 1 + 2.63e10T + 7.25e24T^{2} \)
31 \( 1 - 3.03e12T + 2.25e25T^{2} \)
37 \( 1 - 8.44e12iT - 4.56e26T^{2} \)
41 \( 1 - 5.70e13T + 2.61e27T^{2} \)
43 \( 1 + 6.40e13iT - 5.87e27T^{2} \)
47 \( 1 - 2.44e14iT - 2.66e28T^{2} \)
53 \( 1 - 1.01e14iT - 2.05e29T^{2} \)
59 \( 1 + 1.41e15T + 1.27e30T^{2} \)
61 \( 1 - 2.31e15T + 2.24e30T^{2} \)
67 \( 1 - 4.06e15iT - 1.10e31T^{2} \)
71 \( 1 - 2.68e15T + 2.96e31T^{2} \)
73 \( 1 + 6.95e15iT - 4.74e31T^{2} \)
79 \( 1 + 8.28e15T + 1.81e32T^{2} \)
83 \( 1 + 2.66e16iT - 4.21e32T^{2} \)
89 \( 1 - 6.55e15T + 1.37e33T^{2} \)
97 \( 1 - 1.11e17iT - 5.95e33T^{2} \)
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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

−18.98932153241041275282541459119, −17.64820454601214450207649455933, −15.72233440908653828247335170083, −13.18360876978669830528511350823, −12.41957773978635731137679954161, −10.34389067388701929599474772530, −7.85501291042625067932405510476, −5.89468869040389866869857399242, −2.43200733452414992725343338187, −1.11127694271998320005496516243, 2.69952514967163291414011894809, 5.24891551274259288026082385505, 7.21136442477289125728459447530, 9.826126111232688370948262560040, 11.09657099127338490460631812731, 13.98698251569519042176043740467, 15.50488293264383102071253707378, 16.42403590719557727708168479991, 18.24499809588430535568146985115, 20.57885640841167397590635231785

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