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  + 1.51·2-s + 3-s + 0.296·4-s + 1.13·5-s + 1.51·6-s + 2.34·7-s − 2.58·8-s + 9-s + 1.71·10-s + 4.67·11-s + 0.296·12-s + 4.73·13-s + 3.55·14-s + 1.13·15-s − 4.50·16-s + 17-s + 1.51·18-s − 1.43·19-s + 0.336·20-s + 2.34·21-s + 7.09·22-s + 1.75·23-s − 2.58·24-s − 3.71·25-s + 7.18·26-s + 27-s + 0.695·28-s + ⋯
L(s)  = 1  + 1.07·2-s + 0.577·3-s + 0.148·4-s + 0.507·5-s + 0.618·6-s + 0.886·7-s − 0.912·8-s + 0.333·9-s + 0.543·10-s + 1.41·11-s + 0.0855·12-s + 1.31·13-s + 0.950·14-s + 0.292·15-s − 1.12·16-s + 0.242·17-s + 0.357·18-s − 0.329·19-s + 0.0752·20-s + 0.511·21-s + 1.51·22-s + 0.366·23-s − 0.526·24-s − 0.742·25-s + 1.40·26-s + 0.192·27-s + 0.131·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.174634355$
$L(\frac12)$  $\approx$  $5.174634355$
$L(\frac{3}{2})$   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 \( 1 - 1.51T + 2T^{2} \)
5 \( 1 - 1.13T + 5T^{2} \)
7 \( 1 - 2.34T + 7T^{2} \)
11 \( 1 - 4.67T + 11T^{2} \)
13 \( 1 - 4.73T + 13T^{2} \)
19 \( 1 + 1.43T + 19T^{2} \)
23 \( 1 - 1.75T + 23T^{2} \)
29 \( 1 + 10.5T + 29T^{2} \)
31 \( 1 - 3.63T + 31T^{2} \)
37 \( 1 - 6.81T + 37T^{2} \)
41 \( 1 - 1.80T + 41T^{2} \)
43 \( 1 + 4.41T + 43T^{2} \)
47 \( 1 - 7.11T + 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + 9.83T + 59T^{2} \)
61 \( 1 - 7.86T + 61T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 + 0.981T + 71T^{2} \)
73 \( 1 + 8.45T + 73T^{2} \)
83 \( 1 + 0.0563T + 83T^{2} \)
89 \( 1 + 4.99T + 89T^{2} \)
97 \( 1 + 5.59T + 97T^{2} \)
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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

−8.579949577965290638411409337099, −7.72302292124058195451042454768, −6.76785644858871390372846259107, −5.97263301955874523403401054206, −5.55193905062873077632779741353, −4.37288901341984089804649407346, −4.02157034487732861951622440959, −3.23231864376862219627989335177, −2.08473497023800322561886335409, −1.22388629372659223769787181501, 1.22388629372659223769787181501, 2.08473497023800322561886335409, 3.23231864376862219627989335177, 4.02157034487732861951622440959, 4.37288901341984089804649407346, 5.55193905062873077632779741353, 5.97263301955874523403401054206, 6.76785644858871390372846259107, 7.72302292124058195451042454768, 8.579949577965290638411409337099

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