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  + 1.70·3-s − 0.637·5-s + 1.14·7-s − 0.0945·9-s + 0.485·11-s − 4.68·13-s − 1.08·15-s − 17-s + 1.90·19-s + 1.95·21-s − 0.312·23-s − 4.59·25-s − 5.27·27-s − 3.80·29-s − 1.49·31-s + 0.827·33-s − 0.731·35-s + 3.52·37-s − 7.97·39-s − 5.10·41-s − 2.74·43-s + 0.0603·45-s + 5.63·47-s − 5.68·49-s − 1.70·51-s − 5.67·53-s − 0.309·55-s + ⋯
L(s)  = 1  + 0.984·3-s − 0.285·5-s + 0.433·7-s − 0.0315·9-s + 0.146·11-s − 1.29·13-s − 0.280·15-s − 0.242·17-s + 0.436·19-s + 0.426·21-s − 0.0651·23-s − 0.918·25-s − 1.01·27-s − 0.706·29-s − 0.269·31-s + 0.144·33-s − 0.123·35-s + 0.578·37-s − 1.27·39-s − 0.797·41-s − 0.417·43-s + 0.00899·45-s + 0.821·47-s − 0.811·49-s − 0.238·51-s − 0.779·53-s − 0.0417·55-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 \( 1 - 1.70T + 3T^{2} \)
5 \( 1 + 0.637T + 5T^{2} \)
7 \( 1 - 1.14T + 7T^{2} \)
11 \( 1 - 0.485T + 11T^{2} \)
13 \( 1 + 4.68T + 13T^{2} \)
19 \( 1 - 1.90T + 19T^{2} \)
23 \( 1 + 0.312T + 23T^{2} \)
29 \( 1 + 3.80T + 29T^{2} \)
31 \( 1 + 1.49T + 31T^{2} \)
37 \( 1 - 3.52T + 37T^{2} \)
41 \( 1 + 5.10T + 41T^{2} \)
43 \( 1 + 2.74T + 43T^{2} \)
47 \( 1 - 5.63T + 47T^{2} \)
53 \( 1 + 5.67T + 53T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 - 7.64T + 67T^{2} \)
71 \( 1 - 1.03T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 + 0.162T + 79T^{2} \)
83 \( 1 + 1.63T + 83T^{2} \)
89 \( 1 + 18.8T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−7.993381611676128007587357362692, −7.61390430245834783597757060484, −6.83711501044762969249677074791, −5.77250519680818572952715210519, −5.00420920440566689328517239267, −4.15133523319918230590832948111, −3.33839979234577695578013708132, −2.50266244068807000163940849076, −1.70608737982944304768203963650, 0, 1.70608737982944304768203963650, 2.50266244068807000163940849076, 3.33839979234577695578013708132, 4.15133523319918230590832948111, 5.00420920440566689328517239267, 5.77250519680818572952715210519, 6.83711501044762969249677074791, 7.61390430245834783597757060484, 7.993381611676128007587357362692

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