Properties

Degree 2
Conductor $ 97 \cdot 1031 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s − 4-s + 2·5-s − 3·6-s + 4·7-s − 3·8-s + 6·9-s + 2·10-s + 5·11-s + 3·12-s − 2·13-s + 4·14-s − 6·15-s − 16-s + 2·17-s + 6·18-s + 7·19-s − 2·20-s − 12·21-s + 5·22-s + 6·23-s + 9·24-s − 25-s − 2·26-s − 9·27-s − 4·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s − 1/2·4-s + 0.894·5-s − 1.22·6-s + 1.51·7-s − 1.06·8-s + 2·9-s + 0.632·10-s + 1.50·11-s + 0.866·12-s − 0.554·13-s + 1.06·14-s − 1.54·15-s − 1/4·16-s + 0.485·17-s + 1.41·18-s + 1.60·19-s − 0.447·20-s − 2.61·21-s + 1.06·22-s + 1.25·23-s + 1.83·24-s − 1/5·25-s − 0.392·26-s − 1.73·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100007\)    =    \(97 \cdot 1031\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{100007} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 100007,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.318947284$
$L(\frac12)$  $\approx$  $3.318947284$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{97,\;1031\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{97,\;1031\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad97 \( 1 + T \)
1031 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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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

−13.76145828676657, −13.35455489310518, −12.59070614360823, −12.18387311700897, −11.82369380355072, −11.48276585462297, −11.12317593414348, −10.18962699284075, −10.04014265715410, −9.429097334983883, −8.883686302832510, −8.325656787741897, −7.516429623269297, −6.912434494241635, −6.573549738496345, −5.822375525939673, −5.424832630412492, −5.194641805293854, −4.595877105201048, −4.285477085192517, −3.458395289338638, −2.693613632595304, −1.566759747140602, −1.261139612450317, −0.6604728009988034, 0.6604728009988034, 1.261139612450317, 1.566759747140602, 2.693613632595304, 3.458395289338638, 4.285477085192517, 4.595877105201048, 5.194641805293854, 5.424832630412492, 5.822375525939673, 6.573549738496345, 6.912434494241635, 7.516429623269297, 8.325656787741897, 8.883686302832510, 9.429097334983883, 10.04014265715410, 10.18962699284075, 11.12317593414348, 11.48276585462297, 11.82369380355072, 12.18387311700897, 12.59070614360823, 13.35455489310518, 13.76145828676657

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