Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 17^{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 − 5-s + 6-s + 7-s + 3·8-s + 9-s + 10-s − 4·11-s + 12-s + 6·13-s − 14-s + 15-s − 16-s − 18-s − 4·19-s + 20-s − 21-s + 4·22-s − 3·24-s + 25-s − 6·26-s − 27-s − 28-s − 6·29-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 + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 1.66·13-s − 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.218·21-s + 0.852·22-s − 0.612·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(30345\)    =    \(3 \cdot 5 \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{30345} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 30345,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7508167422$
$L(\frac12)$  $\approx$  $0.7508167422$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
17 \( 1 \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−15.37164623208285, −14.54835926706153, −14.09564149407373, −13.33215818312697, −12.94315329138845, −12.73781828374782, −11.62013774525291, −11.29172830981538, −10.80926875868081, −10.30795609493535, −9.879749852679959, −8.983317141195604, −8.592239446431528, −8.090010912487383, −7.633381921217452, −7.006765957900467, −6.105052448738177, −5.725227028954528, −4.937115927560932, −4.342119262797654, −3.917299463454984, −2.979316798173858, −2.001310751573485, −1.166691964960883, −0.4518844959513270, 0.4518844959513270, 1.166691964960883, 2.001310751573485, 2.979316798173858, 3.917299463454984, 4.342119262797654, 4.937115927560932, 5.725227028954528, 6.105052448738177, 7.006765957900467, 7.633381921217452, 8.090010912487383, 8.592239446431528, 8.983317141195604, 9.879749852679959, 10.30795609493535, 10.80926875868081, 11.29172830981538, 11.62013774525291, 12.73781828374782, 12.94315329138845, 13.33215818312697, 14.09564149407373, 14.54835926706153, 15.37164623208285

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