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Properties

Label 2-177870-1.1-c1-0-123
Degree $2$
Conductor $177870$
Sign $1$
Analytic cond. $1420.29$
Root an. cond. $37.6868$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 4·13-s − 15-s + 16-s − 6·17-s + 18-s + 4·19-s − 20-s + 3·23-s + 24-s + 25-s + 4·26-s + 27-s + 9·29-s − 30-s + 2·31-s + 32-s − 6·34-s + 36-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 + 1.10·13-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 1.67·29-s − 0.182·30-s + 0.359·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-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

Degree: \(2\)
Conductor: \(177870\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1420.29\)
Root analytic conductor: \(37.6868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177870,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.928997207\)
\(L(\frac12)\) \(\approx\) \(6.928997207\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
11 \( 1 \)
good13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.27965909557945, −12.71474831917868, −12.47325069906988, −11.56301067791807, −11.46087759812973, −10.98068510845970, −10.42726533361009, −9.880794025643032, −9.299220354165389, −8.800092946522708, −8.320202887771017, −7.979878769969402, −7.311327520806746, −6.756737196817552, −6.463039956831491, −5.899635642420089, −5.075963318023975, −4.773470565088407, −4.129189004391679, −3.709999389682628, −3.154493249910892, −2.585209025481553, −2.098641957206207, −1.151087119538679, −0.7177548193768717, 0.7177548193768717, 1.151087119538679, 2.098641957206207, 2.585209025481553, 3.154493249910892, 3.709999389682628, 4.129189004391679, 4.773470565088407, 5.075963318023975, 5.899635642420089, 6.463039956831491, 6.756737196817552, 7.311327520806746, 7.979878769969402, 8.320202887771017, 8.800092946522708, 9.299220354165389, 9.880794025643032, 10.42726533361009, 10.98068510845970, 11.46087759812973, 11.56301067791807, 12.47325069906988, 12.71474831917868, 13.27965909557945

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