Properties

Degree 2
Conductor 5077
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.45·2-s + 1.58·3-s + 4.05·4-s − 3.55·5-s − 3.90·6-s − 1.72·7-s − 5.04·8-s − 0.477·9-s + 8.73·10-s + 0.521·11-s + 6.43·12-s − 6.48·13-s + 4.24·14-s − 5.63·15-s + 4.30·16-s − 0.0823·17-s + 1.17·18-s − 1.33·19-s − 14.3·20-s − 2.74·21-s − 1.28·22-s + 1.60·23-s − 8.01·24-s + 7.61·25-s + 15.9·26-s − 5.52·27-s − 6.99·28-s + ⋯
L(s)  = 1  − 1.73·2-s + 0.916·3-s + 2.02·4-s − 1.58·5-s − 1.59·6-s − 0.652·7-s − 1.78·8-s − 0.159·9-s + 2.76·10-s + 0.157·11-s + 1.85·12-s − 1.79·13-s + 1.13·14-s − 1.45·15-s + 1.07·16-s − 0.0199·17-s + 0.277·18-s − 0.305·19-s − 3.21·20-s − 0.598·21-s − 0.273·22-s + 0.335·23-s − 1.63·24-s + 1.52·25-s + 3.13·26-s − 1.06·27-s − 1.32·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5077\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5077} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 5077,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.08436350935$
$L(\frac12)$  $\approx$  $0.08436350935$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5077$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 5077$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5077 \( 1+O(T) \)
good2 \( 1 + 2.45T + 2T^{2} \)
3 \( 1 - 1.58T + 3T^{2} \)
5 \( 1 + 3.55T + 5T^{2} \)
7 \( 1 + 1.72T + 7T^{2} \)
11 \( 1 - 0.521T + 11T^{2} \)
13 \( 1 + 6.48T + 13T^{2} \)
17 \( 1 + 0.0823T + 17T^{2} \)
19 \( 1 + 1.33T + 19T^{2} \)
23 \( 1 - 1.60T + 23T^{2} \)
29 \( 1 + 5.29T + 29T^{2} \)
31 \( 1 + 3.36T + 31T^{2} \)
37 \( 1 - 1.70T + 37T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 - 0.681T + 47T^{2} \)
53 \( 1 + 5.03T + 53T^{2} \)
59 \( 1 - 3.94T + 59T^{2} \)
61 \( 1 - 4.73T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 3.21T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 - 8.37T + 83T^{2} \)
89 \( 1 - 5.06T + 89T^{2} \)
97 \( 1 + 9.06T + 97T^{2} \)
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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

−8.113992731577510558275389862433, −7.915446978229926872865696640792, −7.00615291994394067502165105598, −6.85869669058855305395261058452, −5.38309510587203931695005671105, −4.26920390487367769286386673385, −3.33296929733516363652261949484, −2.75712617516844992772617836128, −1.79469421421637012349279387168, −0.18433338596622281648753936448, 0.18433338596622281648753936448, 1.79469421421637012349279387168, 2.75712617516844992772617836128, 3.33296929733516363652261949484, 4.26920390487367769286386673385, 5.38309510587203931695005671105, 6.85869669058855305395261058452, 7.00615291994394067502165105598, 7.915446978229926872865696640792, 8.113992731577510558275389862433

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