Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 23^{2} $
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-s + 4-s − 6-s − 2·7-s − 8-s + 9-s + 11-s + 12-s − 4·13-s + 2·14-s + 16-s + 6·17-s − 18-s + 4·19-s − 2·21-s − 22-s − 24-s − 5·25-s + 4·26-s + 27-s − 2·28-s + 6·29-s + 8·31-s − 32-s + 33-s − 6·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.436·21-s − 0.213·22-s − 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s − 0.377·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.174·33-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 34914 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34914 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(34914\)    =    \(2 \cdot 3 \cdot 11 \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{34914} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 34914,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.956098961$
$L(\frac12)$  $\approx$  $1.956098961$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
11 \( 1 - T \)
23 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−14.89989731396445, −14.53098897848155, −13.93780621991243, −13.49006780813903, −12.71716578916233, −12.30176889582544, −11.76135377198461, −11.38260021350250, −10.32290779301189, −9.980220000237730, −9.632655339258889, −9.298223360515450, −8.395077401363843, −7.808697426935785, −7.675193690978095, −6.790801145373805, −6.353932871784592, −5.675997873832242, −4.899634268417579, −4.227332446515902, −3.253754531934338, −3.008963939197816, −2.260730832792019, −1.313169917157514, −0.6056379614411854, 0.6056379614411854, 1.313169917157514, 2.260730832792019, 3.008963939197816, 3.253754531934338, 4.227332446515902, 4.899634268417579, 5.675997873832242, 6.353932871784592, 6.790801145373805, 7.675193690978095, 7.808697426935785, 8.395077401363843, 9.298223360515450, 9.632655339258889, 9.980220000237730, 10.32290779301189, 11.38260021350250, 11.76135377198461, 12.30176889582544, 12.71716578916233, 13.49006780813903, 13.93780621991243, 14.53098897848155, 14.89989731396445

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