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  + 0.0509·2-s + 3-s − 1.99·4-s + 3.95·5-s + 0.0509·6-s + 1.89·7-s − 0.203·8-s + 9-s + 0.201·10-s + 5.26·11-s − 1.99·12-s − 2.15·13-s + 0.0963·14-s + 3.95·15-s + 3.98·16-s + 17-s + 0.0509·18-s + 3.28·19-s − 7.90·20-s + 1.89·21-s + 0.268·22-s − 2.60·23-s − 0.203·24-s + 10.6·25-s − 0.109·26-s + 27-s − 3.78·28-s + ⋯
L(s)  = 1  + 0.0360·2-s + 0.577·3-s − 0.998·4-s + 1.77·5-s + 0.0207·6-s + 0.715·7-s − 0.0719·8-s + 0.333·9-s + 0.0637·10-s + 1.58·11-s − 0.576·12-s − 0.597·13-s + 0.0257·14-s + 1.02·15-s + 0.996·16-s + 0.242·17-s + 0.0120·18-s + 0.753·19-s − 1.76·20-s + 0.413·21-s + 0.0571·22-s − 0.542·23-s − 0.0415·24-s + 2.13·25-s − 0.0215·26-s + 0.192·27-s − 0.714·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.410683098$
$L(\frac12)$  $\approx$  $3.410683098$
$L(\frac{3}{2})$   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 \( 1 - 0.0509T + 2T^{2} \)
5 \( 1 - 3.95T + 5T^{2} \)
7 \( 1 - 1.89T + 7T^{2} \)
11 \( 1 - 5.26T + 11T^{2} \)
13 \( 1 + 2.15T + 13T^{2} \)
19 \( 1 - 3.28T + 19T^{2} \)
23 \( 1 + 2.60T + 23T^{2} \)
29 \( 1 - 1.79T + 29T^{2} \)
31 \( 1 - 1.06T + 31T^{2} \)
37 \( 1 + 6.27T + 37T^{2} \)
41 \( 1 - 8.76T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 - 6.79T + 47T^{2} \)
53 \( 1 + 6.84T + 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 + 0.635T + 71T^{2} \)
73 \( 1 + 3.45T + 73T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + 6.91T + 89T^{2} \)
97 \( 1 - 2.23T + 97T^{2} \)
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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

−8.693213940711212893080143703932, −7.88587938832406321555068793832, −6.91644975958720740256722511401, −6.14973382376852510127178553228, −5.37286612157629890961086029520, −4.75972067763705045183844944856, −3.90383521402293004055143394381, −2.91257986476183056780436559514, −1.79394432209538092529775278829, −1.19070889481419289845903346318, 1.19070889481419289845903346318, 1.79394432209538092529775278829, 2.91257986476183056780436559514, 3.90383521402293004055143394381, 4.75972067763705045183844944856, 5.37286612157629890961086029520, 6.14973382376852510127178553228, 6.91644975958720740256722511401, 7.88587938832406321555068793832, 8.693213940711212893080143703932

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