Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11^{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 + 8-s + 9-s − 10-s − 12-s + 2·13-s + 15-s + 16-s − 6·17-s + 18-s + 8·19-s − 20-s − 24-s + 25-s + 2·26-s − 27-s − 6·29-s + 30-s + 4·31-s + 32-s − 6·34-s + 36-s − 10·37-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 − 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 1.83·19-s − 0.223·20-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 1.11·29-s + 0.182·30-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 1.64·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(177870\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{177870} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 177870,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.145491236\)
\(L(\frac12)\)  \(\approx\)  \(2.145491236\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.14661055796713, −12.81722839925281, −12.01001122549248, −11.82755127926042, −11.49269727824872, −10.80219007582895, −10.66813321636032, −9.956265225141519, −9.357699278920096, −8.928215530955004, −8.322921853783086, −7.724158549945284, −7.250738560097092, −6.816238965488735, −6.333555356001351, −5.794993175702337, −5.119361345254828, −4.981808534936368, −4.197257284821990, −3.735111676913006, −3.259565723926460, −2.594733902237290, −1.807329228929584, −1.266820990888011, −0.3932548098038418, 0.3932548098038418, 1.266820990888011, 1.807329228929584, 2.594733902237290, 3.259565723926460, 3.735111676913006, 4.197257284821990, 4.981808534936368, 5.119361345254828, 5.794993175702337, 6.333555356001351, 6.816238965488735, 7.250738560097092, 7.724158549945284, 8.322921853783086, 8.928215530955004, 9.357699278920096, 9.956265225141519, 10.66813321636032, 10.80219007582895, 11.49269727824872, 11.82755127926042, 12.01001122549248, 12.81722839925281, 13.14661055796713

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