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  + 1.18·3-s + 0.677·5-s + 1.46·7-s − 1.58·9-s − 2.15·11-s + 4.65·13-s + 0.805·15-s − 17-s − 6.39·19-s + 1.73·21-s − 8.06·23-s − 4.54·25-s − 5.45·27-s − 0.503·29-s − 6.46·31-s − 2.56·33-s + 0.990·35-s − 4.81·37-s + 5.53·39-s + 3.96·41-s − 3.76·43-s − 1.07·45-s − 5.61·47-s − 4.86·49-s − 1.18·51-s + 5.58·53-s − 1.46·55-s + ⋯
L(s)  = 1  + 0.686·3-s + 0.303·5-s + 0.552·7-s − 0.529·9-s − 0.650·11-s + 1.29·13-s + 0.207·15-s − 0.242·17-s − 1.46·19-s + 0.379·21-s − 1.68·23-s − 0.908·25-s − 1.04·27-s − 0.0934·29-s − 1.16·31-s − 0.446·33-s + 0.167·35-s − 0.791·37-s + 0.886·39-s + 0.618·41-s − 0.573·43-s − 0.160·45-s − 0.818·47-s − 0.694·49-s − 0.166·51-s + 0.767·53-s − 0.197·55-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 \( 1 - 1.18T + 3T^{2} \)
5 \( 1 - 0.677T + 5T^{2} \)
7 \( 1 - 1.46T + 7T^{2} \)
11 \( 1 + 2.15T + 11T^{2} \)
13 \( 1 - 4.65T + 13T^{2} \)
19 \( 1 + 6.39T + 19T^{2} \)
23 \( 1 + 8.06T + 23T^{2} \)
29 \( 1 + 0.503T + 29T^{2} \)
31 \( 1 + 6.46T + 31T^{2} \)
37 \( 1 + 4.81T + 37T^{2} \)
41 \( 1 - 3.96T + 41T^{2} \)
43 \( 1 + 3.76T + 43T^{2} \)
47 \( 1 + 5.61T + 47T^{2} \)
53 \( 1 - 5.58T + 53T^{2} \)
61 \( 1 + 2.84T + 61T^{2} \)
67 \( 1 - 3.63T + 67T^{2} \)
71 \( 1 - 6.38T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 1.34T + 83T^{2} \)
89 \( 1 - 18.7T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−8.213551354584324434017323561301, −7.65461114342850483389765683276, −6.45826207189348241013924864654, −5.92393846735680841162233234502, −5.14942257567053977176850900243, −4.03983670028627472369336426224, −3.50643754860704849601225970021, −2.25723290941815816973851536964, −1.81665253346831263061175210437, 0, 1.81665253346831263061175210437, 2.25723290941815816973851536964, 3.50643754860704849601225970021, 4.03983670028627472369336426224, 5.14942257567053977176850900243, 5.92393846735680841162233234502, 6.45826207189348241013924864654, 7.65461114342850483389765683276, 8.213551354584324434017323561301

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