Properties

Degree 2
Conductor $ 2^{2} \cdot 19 \cdot 53 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.97·3-s + 0.840·5-s − 4.22·7-s + 5.85·9-s + 3.71·11-s − 1.07·13-s − 2.50·15-s + 1.37·17-s − 19-s + 12.5·21-s − 4.51·23-s − 4.29·25-s − 8.48·27-s + 0.356·29-s − 4.24·31-s − 11.0·33-s − 3.54·35-s + 4.55·37-s + 3.18·39-s + 10.9·41-s + 5.17·43-s + 4.91·45-s − 4.68·47-s + 10.8·49-s − 4.09·51-s + 53-s + 3.12·55-s + ⋯
L(s)  = 1  − 1.71·3-s + 0.375·5-s − 1.59·7-s + 1.95·9-s + 1.12·11-s − 0.296·13-s − 0.645·15-s + 0.333·17-s − 0.229·19-s + 2.74·21-s − 0.941·23-s − 0.858·25-s − 1.63·27-s + 0.0662·29-s − 0.763·31-s − 1.92·33-s − 0.599·35-s + 0.749·37-s + 0.509·39-s + 1.71·41-s + 0.789·43-s + 0.733·45-s − 0.683·47-s + 1.54·49-s − 0.572·51-s + 0.137·53-s + 0.421·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4028\)    =    \(2^{2} \cdot 19 \cdot 53\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4028} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4028,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;53\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;53\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
19 \( 1 + T \)
53 \( 1 - T \)
good3 \( 1 + 2.97T + 3T^{2} \)
5 \( 1 - 0.840T + 5T^{2} \)
7 \( 1 + 4.22T + 7T^{2} \)
11 \( 1 - 3.71T + 11T^{2} \)
13 \( 1 + 1.07T + 13T^{2} \)
17 \( 1 - 1.37T + 17T^{2} \)
23 \( 1 + 4.51T + 23T^{2} \)
29 \( 1 - 0.356T + 29T^{2} \)
31 \( 1 + 4.24T + 31T^{2} \)
37 \( 1 - 4.55T + 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 - 5.17T + 43T^{2} \)
47 \( 1 + 4.68T + 47T^{2} \)
59 \( 1 - 7.35T + 59T^{2} \)
61 \( 1 - 6.24T + 61T^{2} \)
67 \( 1 - 3.54T + 67T^{2} \)
71 \( 1 + 5.04T + 71T^{2} \)
73 \( 1 - 5.95T + 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + 3.23T + 83T^{2} \)
89 \( 1 + 0.365T + 89T^{2} \)
97 \( 1 + 4.23T + 97T^{2} \)
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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

−7.84992365917008550012648113800, −6.95875383111751815314854870296, −6.42918736166815419783196496637, −5.95169830539232132833040998741, −5.41222288148006338930809174909, −4.22674616773108368194116278376, −3.72521336419596382515975523680, −2.35121928740468420458617227832, −1.05065915857955406434796422669, 0, 1.05065915857955406434796422669, 2.35121928740468420458617227832, 3.72521336419596382515975523680, 4.22674616773108368194116278376, 5.41222288148006338930809174909, 5.95169830539232132833040998741, 6.42918736166815419783196496637, 6.95875383111751815314854870296, 7.84992365917008550012648113800

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