Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 7^{2} \cdot 13 \cdot 17 $
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 − 5-s + 6-s + 8-s + 9-s − 10-s − 2·11-s + 12-s − 13-s − 15-s + 16-s + 17-s + 18-s − 19-s − 20-s − 2·22-s + 3·23-s + 24-s − 4·25-s − 26-s + 27-s + 2·29-s − 30-s + 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.426·22-s + 0.625·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.371·29-s − 0.182·30-s + 0.718·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 64974 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 64974 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(64974\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{64974} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 64974,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7,\;13,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;13,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 \)
13 \( 1 + T \)
17 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + T + p T^{2} \)
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

−14.43883262041802, −13.97724271129764, −13.51897603445895, −13.01890630559313, −12.47880939440680, −12.13436228540112, −11.54503502118419, −10.96188220527029, −10.50764258362855, −9.912661465833109, −9.458319525368783, −8.668630037308331, −8.260457821005130, −7.693412564401107, −7.282080209288509, −6.627640175977209, −6.148786031688433, −5.149869750237112, −5.114927291348908, −4.200615927754002, −3.752755917017275, −3.121581272246496, −2.559453557299025, −1.938842419460805, −1.067022917138527, 0, 1.067022917138527, 1.938842419460805, 2.559453557299025, 3.121581272246496, 3.752755917017275, 4.200615927754002, 5.114927291348908, 5.149869750237112, 6.148786031688433, 6.627640175977209, 7.282080209288509, 7.693412564401107, 8.260457821005130, 8.668630037308331, 9.458319525368783, 9.912661465833109, 10.50764258362855, 10.96188220527029, 11.54503502118419, 12.13436228540112, 12.47880939440680, 13.01890630559313, 13.51897603445895, 13.97724271129764, 14.43883262041802

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