Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 13^{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 − 3-s + 4-s − 5-s + 6-s + 4·7-s − 8-s + 9-s + 10-s + 4·11-s − 12-s − 4·14-s + 15-s + 16-s − 17-s − 18-s + 2·19-s − 20-s − 4·21-s − 4·22-s − 4·23-s + 24-s + 25-s − 27-s + 4·28-s − 4·29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s − 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.872·21-s − 0.852·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.755·28-s − 0.742·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(86190\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{86190} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 86190,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.975432811$
$L(\frac12)$  $\approx$  $1.975432811$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 \)
17 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−14.13072534580786, −13.45727703932809, −12.62351316568613, −12.33043484790563, −11.68897679650772, −11.33845954576592, −11.10132743660057, −10.64159548044066, −9.821446775845404, −9.357408853280812, −9.018323405666540, −8.204487823740131, −7.809767425619774, −7.580597645079526, −6.763625536265782, −6.320063417386590, −5.711888015629348, −5.144168922741174, −4.364339329453102, −4.152863538324725, −3.367139600556522, −2.373026217304116, −1.819089780272419, −1.137678201029365, −0.6090763393391888, 0.6090763393391888, 1.137678201029365, 1.819089780272419, 2.373026217304116, 3.367139600556522, 4.152863538324725, 4.364339329453102, 5.144168922741174, 5.711888015629348, 6.320063417386590, 6.763625536265782, 7.580597645079526, 7.809767425619774, 8.204487823740131, 9.018323405666540, 9.357408853280812, 9.821446775845404, 10.64159548044066, 11.10132743660057, 11.33845954576592, 11.68897679650772, 12.33043484790563, 12.62351316568613, 13.45727703932809, 14.13072534580786

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