Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 11 \cdot 13^{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 + 2·5-s − 6-s − 7-s − 3·8-s + 9-s + 2·10-s + 11-s + 12-s − 14-s − 2·15-s − 16-s + 2·17-s + 18-s − 4·19-s − 2·20-s + 21-s + 22-s + 3·24-s − 25-s − 27-s + 28-s − 2·29-s − 2·30-s − 8·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.288·12-s − 0.267·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.218·21-s + 0.213·22-s + 0.612·24-s − 1/5·25-s − 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.365·30-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(39039\)    =    \(3 \cdot 7 \cdot 11 \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{39039} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 39039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.291982983$
$L(\frac12)$  $\approx$  $1.291982983$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.73874111834352, −14.11553622457323, −13.74858841536274, −13.30522648058171, −12.72785234560153, −12.37409540928564, −11.89367953393276, −11.19970426022584, −10.55641181570562, −10.06855634095704, −9.587047465639883, −9.010358305737277, −8.607219030381330, −7.775833372511575, −6.971147453379092, −6.490238095311588, −5.939148777422289, −5.393065543947127, −5.118192211000862, −4.200091351597087, −3.747954834923878, −3.104724080328475, −2.150952817983500, −1.526216848255069, −0.3794223528954751, 0.3794223528954751, 1.526216848255069, 2.150952817983500, 3.104724080328475, 3.747954834923878, 4.200091351597087, 5.118192211000862, 5.393065543947127, 5.939148777422289, 6.490238095311588, 6.971147453379092, 7.775833372511575, 8.607219030381330, 9.010358305737277, 9.587047465639883, 10.06855634095704, 10.55641181570562, 11.19970426022584, 11.89367953393276, 12.37409540928564, 12.72785234560153, 13.30522648058171, 13.74858841536274, 14.11553622457323, 14.73874111834352

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