Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.43·3-s − 0.659·5-s + 4.52·7-s + 2.91·9-s + 0.926·11-s + 3.23·13-s + 1.60·15-s − 17-s − 5.97·19-s − 11.0·21-s + 1.60·23-s − 4.56·25-s + 0.212·27-s − 9.97·29-s − 9.56·31-s − 2.25·33-s − 2.98·35-s + 7.27·37-s − 7.85·39-s − 5.10·41-s + 3.90·43-s − 1.92·45-s − 10.6·47-s + 13.4·49-s + 2.43·51-s + 6.72·53-s − 0.611·55-s + ⋯
L(s)  = 1  − 1.40·3-s − 0.294·5-s + 1.70·7-s + 0.970·9-s + 0.279·11-s + 0.896·13-s + 0.414·15-s − 0.242·17-s − 1.37·19-s − 2.40·21-s + 0.335·23-s − 0.913·25-s + 0.0408·27-s − 1.85·29-s − 1.71·31-s − 0.392·33-s − 0.504·35-s + 1.19·37-s − 1.25·39-s − 0.796·41-s + 0.595·43-s − 0.286·45-s − 1.55·47-s + 1.92·49-s + 0.340·51-s + 0.923·53-s − 0.0823·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17,\;59\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
17 \( 1 + T \)
59 \( 1 - T \)
good3 \( 1 + 2.43T + 3T^{2} \)
5 \( 1 + 0.659T + 5T^{2} \)
7 \( 1 - 4.52T + 7T^{2} \)
11 \( 1 - 0.926T + 11T^{2} \)
13 \( 1 - 3.23T + 13T^{2} \)
19 \( 1 + 5.97T + 19T^{2} \)
23 \( 1 - 1.60T + 23T^{2} \)
29 \( 1 + 9.97T + 29T^{2} \)
31 \( 1 + 9.56T + 31T^{2} \)
37 \( 1 - 7.27T + 37T^{2} \)
41 \( 1 + 5.10T + 41T^{2} \)
43 \( 1 - 3.90T + 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 - 6.72T + 53T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 - 8.56T + 67T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 - 1.91T + 79T^{2} \)
83 \( 1 + 7.51T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 8.67T + 97T^{2} \)
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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

−8.098343782550151681583247727019, −7.27010841331107666679280958885, −6.53354829494446063233230435148, −5.63082749575157247322113508172, −5.31345204227415164201852351943, −4.29160307302116622572221894255, −3.87140084478920249649673271123, −2.09277038921347771377113420179, −1.33433004256844929769903933298, 0, 1.33433004256844929769903933298, 2.09277038921347771377113420179, 3.87140084478920249649673271123, 4.29160307302116622572221894255, 5.31345204227415164201852351943, 5.63082749575157247322113508172, 6.53354829494446063233230435148, 7.27010841331107666679280958885, 8.098343782550151681583247727019

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