Properties

Degree 2
Conductor $ 2^{2} \cdot 17 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 9-s − 2·13-s + 16-s + 17-s + 18-s + 25-s + 2·26-s − 32-s − 34-s − 36-s − 49-s − 50-s − 2·52-s + 2·53-s + 64-s + 68-s + 72-s + 81-s − 2·89-s + 98-s + 100-s − 2·101-s + 2·104-s − 2·106-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s − 9-s − 2·13-s + 16-s + 17-s + 18-s + 25-s + 2·26-s − 32-s − 34-s − 36-s − 49-s − 50-s − 2·52-s + 2·53-s + 64-s + 68-s + 72-s + 81-s − 2·89-s + 98-s + 100-s − 2·101-s + 2·104-s − 2·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(68\)    =    \(2^{2} \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{68} (67, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 68,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.3327885867\)
\(L(\frac12)\)  \(\approx\)  \(0.3327885867\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;17\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
17 \( 1 - T \)
good3 \( 1 + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( ( 1 + T )^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 + T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.02449061344557603691883920300, −14.36118879225983677422230504881, −12.47090892005841483622787798176, −11.65059393536294409108661621379, −10.36498251692829322753123217545, −9.378387468381204143197364963903, −8.141436839216771787296709142564, −7.01903598371148299622349237377, −5.41265195424507937916978031210, −2.75562279580887633473300699062, 2.75562279580887633473300699062, 5.41265195424507937916978031210, 7.01903598371148299622349237377, 8.141436839216771787296709142564, 9.378387468381204143197364963903, 10.36498251692829322753123217545, 11.65059393536294409108661621379, 12.47090892005841483622787798176, 14.36118879225983677422230504881, 15.02449061344557603691883920300

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