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  − 3.27·3-s + 3.82·5-s − 1.10·7-s + 7.70·9-s + 4.05·11-s + 2.22·13-s − 12.5·15-s − 17-s − 5.58·19-s + 3.60·21-s − 8.79·23-s + 9.60·25-s − 15.3·27-s − 7.67·29-s + 4.75·31-s − 13.2·33-s − 4.20·35-s + 1.29·37-s − 7.27·39-s − 6.85·41-s − 2.65·43-s + 29.4·45-s − 11.7·47-s − 5.78·49-s + 3.27·51-s − 9.59·53-s + 15.4·55-s + ⋯
L(s)  = 1  − 1.88·3-s + 1.70·5-s − 0.415·7-s + 2.56·9-s + 1.22·11-s + 0.617·13-s − 3.22·15-s − 0.242·17-s − 1.28·19-s + 0.785·21-s − 1.83·23-s + 1.92·25-s − 2.95·27-s − 1.42·29-s + 0.854·31-s − 2.30·33-s − 0.710·35-s + 0.213·37-s − 1.16·39-s − 1.07·41-s − 0.405·43-s + 4.38·45-s − 1.70·47-s − 0.826·49-s + 0.458·51-s − 1.31·53-s + 2.08·55-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 \( 1 + 3.27T + 3T^{2} \)
5 \( 1 - 3.82T + 5T^{2} \)
7 \( 1 + 1.10T + 7T^{2} \)
11 \( 1 - 4.05T + 11T^{2} \)
13 \( 1 - 2.22T + 13T^{2} \)
19 \( 1 + 5.58T + 19T^{2} \)
23 \( 1 + 8.79T + 23T^{2} \)
29 \( 1 + 7.67T + 29T^{2} \)
31 \( 1 - 4.75T + 31T^{2} \)
37 \( 1 - 1.29T + 37T^{2} \)
41 \( 1 + 6.85T + 41T^{2} \)
43 \( 1 + 2.65T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 + 9.59T + 53T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 - 2.30T + 67T^{2} \)
71 \( 1 - 6.69T + 71T^{2} \)
73 \( 1 - 9.58T + 73T^{2} \)
79 \( 1 - 5.58T + 79T^{2} \)
83 \( 1 + 8.10T + 83T^{2} \)
89 \( 1 - 8.45T + 89T^{2} \)
97 \( 1 + 3.32T + 97T^{2} \)
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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

−8.036853089570065392119885470304, −6.65177269654539609537899435459, −6.39992782595822110021552195882, −6.12420722710628541557637491767, −5.30253042040905605367410249107, −4.52390572870619788078411027963, −3.66422431408558999955487766107, −1.93279542512458449066619225962, −1.47039421325910029539278827958, 0, 1.47039421325910029539278827958, 1.93279542512458449066619225962, 3.66422431408558999955487766107, 4.52390572870619788078411027963, 5.30253042040905605367410249107, 6.12420722710628541557637491767, 6.39992782595822110021552195882, 6.65177269654539609537899435459, 8.036853089570065392119885470304

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