Properties

Label 2-369600-1.1-c1-0-106
Degree $2$
Conductor $369600$
Sign $1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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  − 3-s − 7-s + 9-s − 11-s + 6·13-s + 5·17-s − 5·19-s + 21-s + 23-s − 27-s − 3·29-s + 33-s − 6·37-s − 6·39-s + 4·41-s − 11·43-s − 4·47-s + 49-s − 5·51-s + 53-s + 5·57-s + 15·59-s + 15·61-s − 63-s + 10·67-s − 69-s − 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s + 1.21·17-s − 1.14·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s − 0.557·29-s + 0.174·33-s − 0.986·37-s − 0.960·39-s + 0.624·41-s − 1.67·43-s − 0.583·47-s + 1/7·49-s − 0.700·51-s + 0.137·53-s + 0.662·57-s + 1.95·59-s + 1.92·61-s − 0.125·63-s + 1.22·67-s − 0.120·69-s − 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.810790839\)
\(L(\frac12)\) \(\approx\) \(1.810790839\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 - 15 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−12.67703046102222, −11.95308098579162, −11.51571762971279, −11.23598667927315, −10.70302928166572, −10.20174290022601, −9.989217215907615, −9.416574637223673, −8.718333166937741, −8.301791950047389, −8.216338077889277, −7.214203044623079, −6.974851752424170, −6.481472271931419, −5.896756914576112, −5.585323489678113, −5.189676053590833, −4.417711579671813, −3.946237450705272, −3.464437295164452, −3.067359748316643, −2.185644877092381, −1.657387667232039, −1.028462416791092, −0.4075549568739809, 0.4075549568739809, 1.028462416791092, 1.657387667232039, 2.185644877092381, 3.067359748316643, 3.464437295164452, 3.946237450705272, 4.417711579671813, 5.189676053590833, 5.585323489678113, 5.896756914576112, 6.481472271931419, 6.974851752424170, 7.214203044623079, 8.216338077889277, 8.301791950047389, 8.718333166937741, 9.416574637223673, 9.989217215907615, 10.20174290022601, 10.70302928166572, 11.23598667927315, 11.51571762971279, 11.95308098579162, 12.67703046102222

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