Properties

Degree $2$
Conductor $177450$
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 + 6-s + 7-s − 8-s + 9-s − 12-s − 14-s + 16-s + 6·17-s − 18-s − 8·19-s − 21-s + 24-s − 27-s + 28-s + 6·29-s + 4·31-s − 32-s − 6·34-s + 36-s − 10·37-s + 8·38-s + 6·41-s + 42-s + 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.83·19-s − 0.218·21-s + 0.204·24-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 1.64·37-s + 1.29·38-s + 0.937·41-s + 0.154·42-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{177450} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.167576737\)
\(L(\frac12)\) \(\approx\) \(1.167576737\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
13 \( 1 \)
good11 \( 1 + p T^{2} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.10719228644337, −12.45993967577108, −12.08393478522133, −11.89705635203143, −11.21006474351652, −10.58206420250314, −10.37736465576034, −10.15078282944118, −9.234056493136232, −8.975400795479222, −8.296642112782681, −7.996574959336069, −7.450565282836390, −6.849417774601253, −6.439188819201147, −5.917364909732471, −5.402963984550109, −4.859973434373720, −4.199810619219487, −3.768990683794891, −2.875309194862183, −2.435462628385528, −1.618838866771842, −1.139532079829350, −0.3969761779474491, 0.3969761779474491, 1.139532079829350, 1.618838866771842, 2.435462628385528, 2.875309194862183, 3.768990683794891, 4.199810619219487, 4.859973434373720, 5.402963984550109, 5.917364909732471, 6.439188819201147, 6.849417774601253, 7.450565282836390, 7.996574959336069, 8.296642112782681, 8.975400795479222, 9.234056493136232, 10.15078282944118, 10.37736465576034, 10.58206420250314, 11.21006474351652, 11.89705635203143, 12.08393478522133, 12.45993967577108, 13.10719228644337

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