Properties

Degree 2
Conductor $ 2 \cdot 17 $
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 − 2·3-s + 4-s − 2·6-s − 4·7-s + 8-s + 9-s + 6·11-s − 2·12-s + 2·13-s − 4·14-s + 16-s − 17-s + 18-s − 4·19-s + 8·21-s + 6·22-s − 2·24-s − 5·25-s + 2·26-s + 4·27-s − 4·28-s − 4·31-s + 32-s − 12·33-s − 34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s + 1.74·21-s + 1.27·22-s − 0.408·24-s − 25-s + 0.392·26-s + 0.769·27-s − 0.755·28-s − 0.718·31-s + 0.176·32-s − 2.08·33-s − 0.171·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 34 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 34 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(34\)    =    \(2 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{34} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 34,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7492772210$
$L(\frac12)$  $\approx$  $0.7492772210$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
17 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−19.92923014104297, −19.14944926131707, −17.44698019761985, −16.63179484427491, −15.73128059955906, −14.21978494242446, −12.85956463289157, −11.95834417367800, −10.84894668716717, −9.286071988264370, −6.653307312611135, −5.999118551719138, −3.902295471226138, 3.902295471226138, 5.999118551719138, 6.653307312611135, 9.286071988264370, 10.84894668716717, 11.95834417367800, 12.85956463289157, 14.21978494242446, 15.73128059955906, 16.63179484427491, 17.44698019761985, 19.14944926131707, 19.92923014104297

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