Properties

Degree 2
Conductor 107
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s − 11-s − 12-s − 13-s + 16-s − 19-s − 23-s + 25-s + 27-s + 2·29-s + 33-s − 37-s + 39-s − 41-s − 44-s + 2·47-s − 48-s + 49-s − 52-s − 53-s + 57-s − 61-s + 64-s + 69-s − 75-s − 76-s + ⋯
L(s)  = 1  − 3-s + 4-s − 11-s − 12-s − 13-s + 16-s − 19-s − 23-s + 25-s + 27-s + 2·29-s + 33-s − 37-s + 39-s − 41-s − 44-s + 2·47-s − 48-s + 49-s − 52-s − 53-s + 57-s − 61-s + 64-s + 69-s − 75-s − 76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(107\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{107} (106, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 107,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5030252953\)
\(L(\frac12)\)  \(\approx\)  \(0.5030252953\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 107$,\(F_p(T)\) is a polynomial of degree 2. If $p = 107$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad107 \( 1 - T \)
good2 \( ( 1 - T )( 1 + T ) \)
3 \( 1 + T + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T + T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( ( 1 - T )^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )^{2} \)
53 \( 1 + T + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )^{2} \)
89 \( 1 + T + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−14.04011337995081042829635922421, −12.41714126767367420427897494631, −12.04504638430051238311412403063, −10.71867685031617401459794955967, −10.31805535325114251598975346150, −8.373724445978545223480806975034, −7.08658080864405272708817928649, −6.07457868538096646079651191674, −4.91802014171830707927308424854, −2.61608028997495214067096557530, 2.61608028997495214067096557530, 4.91802014171830707927308424854, 6.07457868538096646079651191674, 7.08658080864405272708817928649, 8.373724445978545223480806975034, 10.31805535325114251598975346150, 10.71867685031617401459794955967, 12.04504638430051238311412403063, 12.41714126767367420427897494631, 14.04011337995081042829635922421

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