Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 79 \cdot 211 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 2·5-s − 6-s + 2·7-s + 8-s + 9-s − 2·10-s − 6·11-s − 12-s + 13-s + 2·14-s + 2·15-s + 16-s + 3·17-s + 18-s − 6·19-s − 2·20-s − 2·21-s − 6·22-s + 23-s − 24-s − 25-s + 26-s − 27-s + 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.80·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 1.37·19-s − 0.447·20-s − 0.436·21-s − 1.27·22-s + 0.208·23-s − 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.377·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100014\)    =    \(2 \cdot 3 \cdot 79 \cdot 211\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100014} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 100014,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;79,\;211\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;79,\;211\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 + T \)
79 \( 1 - T \)
211 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−13.94366021846652, −13.23899662036929, −13.06303285758476, −12.62103066396797, −12.04463765737934, −11.46702951878261, −11.19147931521195, −10.77856877107009, −10.27764463352681, −9.794127250445113, −8.895220436291807, −8.284013969883161, −7.873691504733430, −7.516270077829493, −7.037423779838183, −6.193850509441654, −5.757172758356286, −5.238902636587102, −4.765357821513146, −4.294795261418342, −3.615959139846408, −3.165628662999580, −2.179561226907387, −1.884665755069365, −0.7410755054625090, 0, 0.7410755054625090, 1.884665755069365, 2.179561226907387, 3.165628662999580, 3.615959139846408, 4.294795261418342, 4.765357821513146, 5.238902636587102, 5.757172758356286, 6.193850509441654, 7.037423779838183, 7.516270077829493, 7.873691504733430, 8.284013969883161, 8.895220436291807, 9.794127250445113, 10.27764463352681, 10.77856877107009, 11.19147931521195, 11.46702951878261, 12.04463765737934, 12.62103066396797, 13.06303285758476, 13.23899662036929, 13.94366021846652

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