Properties

Label 2-177870-1.1-c1-0-25
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 − 2·13-s + 15-s + 16-s − 6·17-s + 18-s + 20-s − 8·23-s + 24-s + 25-s − 2·26-s + 27-s − 10·29-s + 30-s + 8·31-s + 32-s − 6·34-s + 36-s + 2·37-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 + 0.223·20-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 1.85·29-s + 0.182·30-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.328·37-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

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: $\chi_{177870} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177870,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.268026506\)
\(L(\frac12)\) \(\approx\) \(3.268026506\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−13.19083178536566, −12.97060139433110, −12.27055805460594, −11.78113760107240, −11.47136492427271, −10.83327107819521, −10.26354029676544, −9.880943041597199, −9.493593045077403, −8.725839508157021, −8.517695628737866, −7.761333245374858, −7.385456588651511, −6.825671330591147, −6.246311465610942, −5.941596488244587, −5.232483499303648, −4.676648305080073, −4.222809771196869, −3.762939314638564, −3.010048155006362, −2.547608496827474, −1.881682257026109, −1.655473027606417, −0.3936241677039723, 0.3936241677039723, 1.655473027606417, 1.881682257026109, 2.547608496827474, 3.010048155006362, 3.762939314638564, 4.222809771196869, 4.676648305080073, 5.232483499303648, 5.941596488244587, 6.246311465610942, 6.825671330591147, 7.385456588651511, 7.761333245374858, 8.517695628737866, 8.725839508157021, 9.493593045077403, 9.880943041597199, 10.26354029676544, 10.83327107819521, 11.47136492427271, 11.78113760107240, 12.27055805460594, 12.97060139433110, 13.19083178536566

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