Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 79 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.68·2-s + 3-s + 0.825·4-s + 0.800·5-s + 1.68·6-s + 3.61·7-s − 1.97·8-s + 9-s + 1.34·10-s + 4.82·11-s + 0.825·12-s + 0.00709·13-s + 6.06·14-s + 0.800·15-s − 4.96·16-s + 17-s + 1.68·18-s + 4.13·19-s + 0.660·20-s + 3.61·21-s + 8.10·22-s − 0.768·23-s − 1.97·24-s − 4.35·25-s + 0.0119·26-s + 27-s + 2.97·28-s + ⋯
L(s)  = 1  + 1.18·2-s + 0.577·3-s + 0.412·4-s + 0.357·5-s + 0.686·6-s + 1.36·7-s − 0.698·8-s + 0.333·9-s + 0.425·10-s + 1.45·11-s + 0.238·12-s + 0.00196·13-s + 1.62·14-s + 0.206·15-s − 1.24·16-s + 0.242·17-s + 0.396·18-s + 0.948·19-s + 0.147·20-s + 0.787·21-s + 1.72·22-s − 0.160·23-s − 0.403·24-s − 0.871·25-s + 0.00233·26-s + 0.192·27-s + 0.563·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4029\)    =    \(3 \cdot 17 \cdot 79\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4029} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.666258855$
$L(\frac12)$  $\approx$  $5.666258855$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;17,\;79\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;79\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
17 \( 1 - T \)
79 \( 1 - T \)
good2 \( 1 - 1.68T + 2T^{2} \)
5 \( 1 - 0.800T + 5T^{2} \)
7 \( 1 - 3.61T + 7T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 - 0.00709T + 13T^{2} \)
19 \( 1 - 4.13T + 19T^{2} \)
23 \( 1 + 0.768T + 23T^{2} \)
29 \( 1 - 6.53T + 29T^{2} \)
31 \( 1 - 1.44T + 31T^{2} \)
37 \( 1 + 8.76T + 37T^{2} \)
41 \( 1 + 3.87T + 41T^{2} \)
43 \( 1 - 6.04T + 43T^{2} \)
47 \( 1 + 8.36T + 47T^{2} \)
53 \( 1 + 7.19T + 53T^{2} \)
59 \( 1 + 7.87T + 59T^{2} \)
61 \( 1 + 6.09T + 61T^{2} \)
67 \( 1 - 4.26T + 67T^{2} \)
71 \( 1 - 3.62T + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 - 2.78T + 97T^{2} \)
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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

−8.365450362590977086545551158678, −7.77356159561940204446871402486, −6.73954591674690809069921386604, −6.15894402124240620119686288537, −5.15633416143672477977843278511, −4.73782142096641653570260632968, −3.85863760722729217509059045431, −3.25481313425706786046100041466, −2.09983272008989239513932625527, −1.27558022488927570413358658489, 1.27558022488927570413358658489, 2.09983272008989239513932625527, 3.25481313425706786046100041466, 3.85863760722729217509059045431, 4.73782142096641653570260632968, 5.15633416143672477977843278511, 6.15894402124240620119686288537, 6.73954591674690809069921386604, 7.77356159561940204446871402486, 8.365450362590977086545551158678

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