Properties

Label 2-177450-1.1-c1-0-114
Degree $2$
Conductor $177450$
Sign $1$
Analytic cond. $1416.94$
Root an. cond. $37.6423$
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 − 6-s − 7-s − 8-s + 9-s + 4·11-s + 12-s + 14-s + 16-s + 6·17-s − 18-s − 21-s − 4·22-s + 8·23-s − 24-s + 27-s − 28-s + 10·29-s + 8·31-s − 32-s + 4·33-s − 6·34-s + 36-s + 2·37-s + 2·41-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 + 1.20·11-s + 0.288·12-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.218·21-s − 0.852·22-s + 1.66·23-s − 0.204·24-s + 0.192·27-s − 0.188·28-s + 1.85·29-s + 1.43·31-s − 0.176·32-s + 0.696·33-s − 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.312·41-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: $1$
Analytic conductor: \(1416.94\)
Root analytic conductor: \(37.6423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{177450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.556921287\)
\(L(\frac12)\) \(\approx\) \(3.556921287\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
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

−13.10462509888732, −12.62306146566539, −12.19505389714748, −11.77627349268185, −11.29981734042390, −10.70059609167259, −10.08199427044175, −9.881182558910456, −9.375906384673758, −8.832234419560584, −8.518646817378978, −7.963629496970827, −7.476333886034622, −6.910505513407619, −6.460607907911064, −6.139167604925784, −5.310432803572926, −4.701220866062322, −4.204669078916486, −3.362002073940787, −3.052700896890967, −2.632255836930383, −1.600882711600381, −1.164134627275299, −0.6718197599699156, 0.6718197599699156, 1.164134627275299, 1.600882711600381, 2.632255836930383, 3.052700896890967, 3.362002073940787, 4.204669078916486, 4.701220866062322, 5.310432803572926, 6.139167604925784, 6.460607907911064, 6.910505513407619, 7.476333886034622, 7.963629496970827, 8.518646817378978, 8.832234419560584, 9.375906384673758, 9.881182558910456, 10.08199427044175, 10.70059609167259, 11.29981734042390, 11.77627349268185, 12.19505389714748, 12.62306146566539, 13.10462509888732

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