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  + 2.53·3-s − 2.27·5-s + 1.27·7-s + 3.42·9-s + 0.683·11-s − 4.05·13-s − 5.77·15-s + 0.471·17-s − 19-s + 3.23·21-s − 4.28·23-s + 0.195·25-s + 1.07·27-s − 9.77·29-s − 9.37·31-s + 1.73·33-s − 2.91·35-s + 2.27·37-s − 10.2·39-s + 1.23·41-s − 0.166·43-s − 7.80·45-s + 5.60·47-s − 5.36·49-s + 1.19·51-s + 53-s − 1.55·55-s + ⋯
L(s)  = 1  + 1.46·3-s − 1.01·5-s + 0.482·7-s + 1.14·9-s + 0.206·11-s − 1.12·13-s − 1.49·15-s + 0.114·17-s − 0.229·19-s + 0.706·21-s − 0.893·23-s + 0.0390·25-s + 0.206·27-s − 1.81·29-s − 1.68·31-s + 0.301·33-s − 0.492·35-s + 0.374·37-s − 1.64·39-s + 0.192·41-s − 0.0254·43-s − 1.16·45-s + 0.818·47-s − 0.766·49-s + 0.167·51-s + 0.137·53-s − 0.210·55-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 \( 1 - 2.53T + 3T^{2} \)
5 \( 1 + 2.27T + 5T^{2} \)
7 \( 1 - 1.27T + 7T^{2} \)
11 \( 1 - 0.683T + 11T^{2} \)
13 \( 1 + 4.05T + 13T^{2} \)
17 \( 1 - 0.471T + 17T^{2} \)
23 \( 1 + 4.28T + 23T^{2} \)
29 \( 1 + 9.77T + 29T^{2} \)
31 \( 1 + 9.37T + 31T^{2} \)
37 \( 1 - 2.27T + 37T^{2} \)
41 \( 1 - 1.23T + 41T^{2} \)
43 \( 1 + 0.166T + 43T^{2} \)
47 \( 1 - 5.60T + 47T^{2} \)
59 \( 1 + 7.90T + 59T^{2} \)
61 \( 1 - 0.280T + 61T^{2} \)
67 \( 1 - 2.83T + 67T^{2} \)
71 \( 1 - 9.88T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 + 9.28T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + 0.0580T + 97T^{2} \)
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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

−7.87030544752062737654387038864, −7.69227982511058040211147640405, −7.05163802608605341712699611724, −5.79896088504125264704747409954, −4.84721785457797597362516891570, −3.90297109674562027393205153803, −3.60219916661818220521762568600, −2.44933428493403846138748813447, −1.78360180928001586801246394992, 0, 1.78360180928001586801246394992, 2.44933428493403846138748813447, 3.60219916661818220521762568600, 3.90297109674562027393205153803, 4.84721785457797597362516891570, 5.79896088504125264704747409954, 7.05163802608605341712699611724, 7.69227982511058040211147640405, 7.87030544752062737654387038864

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