Properties

Degree 2
Conductor $ 7 \cdot 29 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 7-s − 9-s + 16-s − 2·23-s + 25-s − 28-s − 29-s − 36-s + 49-s + 2·53-s + 63-s + 64-s − 2·67-s + 2·71-s + 81-s − 2·92-s + 100-s + 2·107-s − 2·109-s − 112-s − 116-s + ⋯
L(s)  = 1  + 4-s − 7-s − 9-s + 16-s − 2·23-s + 25-s − 28-s − 29-s − 36-s + 49-s + 2·53-s + 63-s + 64-s − 2·67-s + 2·71-s + 81-s − 2·92-s + 100-s + 2·107-s − 2·109-s − 112-s − 116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(203\)    =    \(7 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{203} (202, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 203,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.7427123261\)
\(L(\frac12)\)  \(\approx\)  \(0.7427123261\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;29\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{7,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 + T \)
29 \( 1 + T \)
good2 \( ( 1 - T )( 1 + T ) \)
3 \( 1 + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( ( 1 + T )^{2} \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T^{2} \)
67 \( ( 1 + T )^{2} \)
71 \( ( 1 - T )^{2} \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−12.45554371406789655146205461029, −11.76290495696752421611164315692, −10.76606916444414979073528745746, −9.871343937942722300828216965189, −8.669450529021041551002595969632, −7.50716075634620640317141126598, −6.40457578835801695383541290378, −5.64393716734289779747085337421, −3.62945398442553331994002713115, −2.43025923380832463517460190752, 2.43025923380832463517460190752, 3.62945398442553331994002713115, 5.64393716734289779747085337421, 6.40457578835801695383541290378, 7.50716075634620640317141126598, 8.669450529021041551002595969632, 9.871343937942722300828216965189, 10.76606916444414979073528745746, 11.76290495696752421611164315692, 12.45554371406789655146205461029

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