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  − 1.88·2-s + 3-s + 1.56·4-s − 0.846·5-s − 1.88·6-s + 3.01·7-s + 0.829·8-s + 9-s + 1.59·10-s − 0.920·11-s + 1.56·12-s − 3.85·13-s − 5.69·14-s − 0.846·15-s − 4.68·16-s + 17-s − 1.88·18-s + 1.27·19-s − 1.32·20-s + 3.01·21-s + 1.73·22-s − 0.718·23-s + 0.829·24-s − 4.28·25-s + 7.27·26-s + 27-s + 4.70·28-s + ⋯
L(s)  = 1  − 1.33·2-s + 0.577·3-s + 0.780·4-s − 0.378·5-s − 0.770·6-s + 1.13·7-s + 0.293·8-s + 0.333·9-s + 0.504·10-s − 0.277·11-s + 0.450·12-s − 1.06·13-s − 1.52·14-s − 0.218·15-s − 1.17·16-s + 0.242·17-s − 0.444·18-s + 0.293·19-s − 0.295·20-s + 0.658·21-s + 0.370·22-s − 0.149·23-s + 0.169·24-s − 0.856·25-s + 1.42·26-s + 0.192·27-s + 0.889·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.075392269$
$L(\frac12)$  $\approx$  $1.075392269$
$L(\frac{3}{2})$   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 \( 1 + 1.88T + 2T^{2} \)
5 \( 1 + 0.846T + 5T^{2} \)
7 \( 1 - 3.01T + 7T^{2} \)
11 \( 1 + 0.920T + 11T^{2} \)
13 \( 1 + 3.85T + 13T^{2} \)
19 \( 1 - 1.27T + 19T^{2} \)
23 \( 1 + 0.718T + 23T^{2} \)
29 \( 1 - 4.88T + 29T^{2} \)
31 \( 1 + 5.70T + 31T^{2} \)
37 \( 1 - 2.37T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 4.21T + 43T^{2} \)
47 \( 1 - 5.34T + 47T^{2} \)
53 \( 1 + 4.27T + 53T^{2} \)
59 \( 1 - 0.996T + 59T^{2} \)
61 \( 1 + 2.60T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 9.80T + 73T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 - 3.60T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−8.210307473049118175323829925056, −7.923919482841696453107865676155, −7.50106701681245049555551274998, −6.65101126264822743889255691651, −5.34124508916378528463524607041, −4.66809422055804379215617274378, −3.82879297043834308698027168818, −2.52746221243727635837883371943, −1.84562702468972293104021140164, −0.71548112325960647166726685221, 0.71548112325960647166726685221, 1.84562702468972293104021140164, 2.52746221243727635837883371943, 3.82879297043834308698027168818, 4.66809422055804379215617274378, 5.34124508916378528463524607041, 6.65101126264822743889255691651, 7.50106701681245049555551274998, 7.923919482841696453107865676155, 8.210307473049118175323829925056

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